ÿþ<html xmlns:v="urn:schemas-microsoft-com:vml" xmlns:o="urn:schemas-microsoft-com:office:office" xmlns:w="urn:schemas-microsoft-com:office:word" xmlns:x="urn:schemas-microsoft-com:office:excel" xmlns="http://www.w3.org/TR/REC-html40"> <head> <meta http-equiv=Content-Type content="text/html; charset=unicode"> <meta name=ProgId content=Word.Document> <meta name=Generator content="Microsoft Word 11"> <meta name=Originator content="Microsoft Word 11"> <link rel=File-List href="pi_converge_files/filelist.xml"> <link rel=Edit-Time-Data href="pi_converge_files/editdata.mso"> <!--[if !mso]> <style> v\:* {behavior:url(#default#VML);} o\:* {behavior:url(#default#VML);} w\:* {behavior:url(#default#VML);} .shape {behavior:url(#default#VML);} </style> <![endif]--> <title>2dCODE-R-past.com &amp; find legitimate facts and clues towards& </title> <!--[if gte mso 9]><xml> <w:WordDocument> <w:Zoom>BestFit</w:Zoom> <w:SpellingState>Clean</w:SpellingState> <w:GrammarState>Clean</w:GrammarState> <w:ValidateAgainstSchemas/> <w:SaveIfXMLInvalid>false</w:SaveIfXMLInvalid> <w:IgnoreMixedContent>false</w:IgnoreMixedContent> <w:AlwaysShowPlaceholderText>false</w:AlwaysShowPlaceholderText> <w:BrowserLevel>MicrosoftInternetExplorer4</w:BrowserLevel> </w:WordDocument> </xml><![endif]--><!--[if gte mso 9]><xml> <w:LatentStyles DefLockedState="false" LatentStyleCount="156"> </w:LatentStyles> </xml><![endif]--> <style> <!-- /* Font Definitions */ @font-face {font-family:"Arial Unicode MS"; 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mso-bidi-font-family:Georgia'>È<span class=SpellE>k<span style='font-family: "Lucida Sans Unicode"'>~</span>eÐd<span style='font-family:Tahoma'>*</span>o</span>Ð]):</span></span><span style='mso-bookmark:OLE_LINK1'></span></a><span style='mso-bookmark:OLE_LINK1'><span style='font-size:11.0pt;font-family:Georgia'>| </span></span><a href="http://www.2dcode-r-past.com/Geometry/index.htm"><span style='mso-bookmark: OLE_LINK1'><span style='font-size:11.0pt;font-family:Georgia'>Sacred Geometry</span></span><span style='mso-bookmark:OLE_LINK1'></span></a><span style='mso-bookmark:OLE_LINK1'><span style='font-size:11.0pt;font-family:Georgia'> | </span></span><a href="http://www.2dcode-r-past.com/Geometry/AE_ratios/Ratios.htm"><span style='mso-bookmark:OLE_LINK1'><span style='font-size:11.0pt;font-family:Georgia'>Ancient Ratios</span></span><span style='mso-bookmark:OLE_LINK1'></span></a><span style='mso-bookmark:OLE_LINK1'><span style='font-size:11.0pt;font-family:Georgia'> | </span></span><a href="http://www.2dcode-r-past.com/links.htm"><span style='mso-bookmark:OLE_LINK1'><span style='font-size:11.0pt;font-family:Georgia'>Links</span></span><span style='mso-bookmark:OLE_LINK1'></span></a><span style='mso-bookmark:OLE_LINK1'><span style='font-size:11.0pt;font-family:Georgia'> |<o:p></o:p></span></span></p> <span style='mso-bookmark:OLE_LINK1'></span> <p class=MsoNormal><span style='font-size:11.0pt;font-family:Georgia'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Converging Constants<span class=GramE>:[</span>by Rob Miller summer/fall 2008][Under analysis]<o:p></o:p></span></b></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Theory: </span></b><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Any value can be approximated via combined </span><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>dual integer sequences creating a pattern of fractional ratios that shrink in deviation and circle or spiral <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:.5in'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'><span style='mso-spacerun:yes'>  </span><span class=GramE>the</span> true input value on both upper and lower boundary limits as they approach infinity and or converge closer and closer towards that input.<span style='color:black'><o:p></o:p></span></span></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p>&nbsp;</o:p></span></b></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>(4)Method(s): </span></b><b style='mso-bidi-font-weight: normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:red'>(2<span class=GramE>)Linear</span>, (1)Concentric &amp; (1)Spiral Convergence <o:p></o:p></span></b></p> <p class=MsoNormal style='margin-left:1.0in;text-indent:.5in'><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:"Courier New";color:black'>[</span></b><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>there</span></span><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'> is no official label to this technique that could be found at the time of this writing<b style='mso-bidi-font-weight:normal'>.]<o:p></o:p></b></span></p> <p class=MsoNormal style='margin-left:1.0in;text-indent:.5in'><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></b></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><span style='mso-spacerun:yes'> </span><span style='mso-tab-count:1'>         </span><span style='mso-spacerun:yes'> </span><span class=GramE><b style='mso-bidi-font-weight: normal'><i style='mso-bidi-font-style:normal'><u>2 methods of  linear convergence currently under analysis.</u></i></b></span><b style='mso-bidi-font-weight: normal'><i style='mso-bidi-font-style:normal'><u><o:p></o:p></u></i></b></span></p> <p class=MsoNormal style='margin-left:1.0in'><span style='font-size:10.0pt; font-family:Verdana'>1.) Lower&nbsp;linear convergence =&nbsp;Upper Limit is set to the Input value so we do not cross above.<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:1.0in'><span style='font-size:10.0pt; font-family:Verdana'>2.) Upper linear convergence =&nbsp;Lower Limit is set to the Input value so we do not cross below.<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:.5in'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:.5in'><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:"Courier New";color:black'>Those two linear techniques naturally combine to create the  spiral technique.<o:p></o:p></span></i></p> <p class=MsoNormal style='margin-left:.5in;text-indent:.5in'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:.5in'><span class=GramE><b style='mso-bidi-font-weight:normal'><i style='mso-bidi-font-style:normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'>2 methods  spiral and  concentric convergence currently under analysis.</span></u></i></b></span><b style='mso-bidi-font-weight:normal'><i style='mso-bidi-font-style:normal'><u><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p></o:p></span></u></i></b></p> <p class=MsoNormal style='margin-left:.5in;text-indent:.5in'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:blue'>3.) Spiral = Independent upper and lower limits where each is converged independently allowing the deviation between constant and upper limit or constant and lower limit.<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><span style='mso-spacerun:yes'>             </span>& More of a</span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:blue'> </span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'>natural lazy spiraling technique& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:.5in'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:red'>4.) <span class=SpellE>Concetric</span> <span class=GramE>= <span style='mso-spacerun:yes'> </span>Forced</span> convergence through mirroring limits, where the upper and lower deviation is mirrored or copied on each hit to ensure a closer and closer value.<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:.5in'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'><span style='mso-spacerun:yes'>    </span>& More concentric in pattern and technique forcing both limits closer each hit& </span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:blue'><o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Output:<o:p></o:p></span></b></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Two integer sequences<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:1.0in;text-indent:-.25in;mso-list:l0 level1 lfo2; tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt; font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:Symbol; color:black'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'>[numerator sequence]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:1.0in;text-indent:-.25in;mso-list:l0 level1 lfo2; tab-stops:list 1.0in'><![if !supportLists]><span style='font-size:10.0pt; font-family:Symbol;mso-fareast-font-family:Symbol;mso-bidi-font-family:Symbol; color:black'><span style='mso-list:Ignore'>·<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'>[denominator sequence] <o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Resulting:</span></b><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'> <o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>One combined Spiral Pattern of converging fractions getting closer to true value as they approach infinity.<span style='mso-spacerun:yes'>          </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New"'><span style='mso-tab-count:1'>            </span><o:p></o:p></span></b></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'><span style='mso-tab-count:1'>            </span></span></b><span style='font-size: 10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>Through analysis, we can see one of the main spiral patterns unfold by comparing the next value of a resulting integer sequence compared to the previous resulting values <o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>in</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'> both numerator and denominator.<span style='mso-spacerun:yes'>  </span><span class=SpellE><span class=GramE><span style='color:red'>numer</span></span></span><span class=GramE><span style='color:red'>(</span></span><span style='color:red'>x)-<span class=SpellE>numer</span>(x-1)</span> along with <span class=SpellE><span style='color:red'>denom</span></span><span style='color:red'>(x)-<span class=SpellE>denom</span>(x-1). <o:p></o:p></span></span></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Finally combine that pattern, in ratio, to see the approximation. [Ex below]<o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New"'><span style='mso-tab-count:1'>          </span><span style='color:red'>(<span class=SpellE><span class=GramE>numer</span></span><span class=GramE>(</span>x)-<span class=SpellE>numer</span>(x-1)) / (<span class=SpellE>denom</span>(x)-<span class=SpellE>denom</span>(x-1)) = ~ input constant</span><span style='color:black'>; and is the proof for this theory and has applied to each constant studied thus far.<o:p></o:p></span></span></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Hence: </span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'><span style='mso-tab-count:1'>         </span><i style='mso-bidi-font-style:normal'><span style='color:black'>** <u>Note or RULE</u>: Through Spiral Convergence; each combined integer sequence has consequently resulted in different generating functions. <o:p></o:p></span></i></span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:.5in'><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:"Courier New";color:black'>However, the concept to proof the resulting combined spiral pattern of convergence has been the same. They all pattern in a combination <o:p></o:p></span></i></p> <p class=MsoNormal style='text-indent:.5in'><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:red'>[<span class=SpellE><span class=GramE>numer</span></span><span class=GramE>(</span>x)-<span class=SpellE>numer</span>(x-1)/<span class=SpellE>denom</span>(x)-<span class=SpellE>denom</span>(x-1)]</span></i><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:red'> = & </span><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:"Courier New";color:black'> An approximation of the very constant that was input.<o:p></o:p></span></i></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>** Interesting enough, all methods appear to generate similar outputs of resulting patterns that comply with an approximation of the input variable. <o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'><o:p>&nbsp;</o:p></span></i></p> <p class=MsoNormal style='margin-left:.5in'><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:green'>** The best example is to compare: <o:p></o:p></span></i></p> <p class=MsoNormal style='margin-left:.5in'><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:green'>Fibonacci Pattern 1,1,2,3,5,8,13,21,34,55,89,144& <span style='mso-tab-count:1'>         </span> {<span class=GramE>sum</span> the last two}<o:p></o:p></span></i></p> <p class=MsoNormal style='margin-left:.5in'><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:green'>Lucas Pattern </span></i><span style='font-size:10.0pt;font-family: Verdana;color:green'>2,1,3,4,7,11,18,29,47,76,123& <span style='mso-tab-count: 2'>                </span></span><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:green'>{<span class=GramE>sum</span> the last two}</span></i><span style='font-size:10.0pt;font-family:Verdana;color:green'><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:green'><o:p>&nbsp;</o:p></span></i></p> <p class=MsoNormal style='margin-left:.5in'><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:green'>the Lucas Pattern </span></i><span style='font-size:10.0pt; font-family:Verdana;color:green'>2,1,3,4,7,11,18,29,47,76,123& </span><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:"Courier New";color:green'> or Fibonacci Pattern viewed as Numerator only (x-(x-1)). <o:p></o:p></span></i></p> <p class=MsoNormal style='text-indent:.5in'><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:green'>The resulting pattern will actually be the Fibonacci Pattern 1,1,2,3,5,8,13,21,34,55,89,144& or Lucas Pattern depending which you choose.<o:p></o:p></span></i></p> <p class=MsoNormal style='margin-left:.5in'><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:green'>Furthermore, The Lucas pattern actually can be defined via </span></i><span class=SpellE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: Arial;color:green'>Fibo_<span class=GramE>Number</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: Arial;color:green'>(</span></span><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:Arial;color:green'>n-3)+<span class=SpellE>Fibo_Number</span>(n-1) = <span class=SpellE>ANY_CURRENT_Lucas_VALUE_N</span> </span><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:"Courier New";color:green'>. <o:p></o:p></span></i></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;color:green'>Observation:<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;color:green'>2Fi1Lu pattern</span></u></b><span style='font-size:10.0pt;font-family:Verdana; color:green'> example starting with the <span class=SpellE>Fibo</span> #  5 triggering the formula& : <b style='mso-bidi-font-weight:normal'> </b></span><span class=SpellE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: Arial;color:green'>Fibo_<span class=GramE>Number</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: Arial;color:green'>(</span></span><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:Arial;color:green'>n-3)+<span class=SpellE>Fibo_Number</span>(n-1)<u> <o:p></o:p></u></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;color:green'><span style='mso-tab-count:3'>                             </span><u>Pattern#1</u><span style='mso-tab-count:1'>      </span>|<span style='mso-tab-count:1'>        </span><u>Pattern#2</u><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:"Courier New";color:green'>Input (Fibo5)<span style='mso-spacerun:yes'>   </span>= Fibo1<span style='mso-tab-count:2'>       </span>+<span style='mso-tab-count:1'>     </span>Fibo3 <span style='mso-tab-count:1'>      </span>= Output Lucas4<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:"Courier New";color:green'>Input (Fibo8)<span style='mso-spacerun:yes'>   </span>= Fibo2<span style='mso-tab-count:2'>       </span>+<span style='mso-tab-count:1'>     </span>Fibo5<span style='mso-tab-count:2'>       </span>= Output Lucas7<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:"Courier New";color:green'>Input (Fibo13<span class=GramE>)<span style='mso-spacerun:yes'>  </span>=</span> Fibo3<span style='mso-tab-count: 2'>       </span>+<span style='mso-tab-count:1'>     </span>Fibo8<span style='mso-tab-count:2'>       </span>= Output Lucas11<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:"Courier New";color:green'>Input (Fibo21<span class=GramE>)<span style='mso-spacerun:yes'>  </span>=</span> Fibo5<span style='mso-tab-count: 2'>       </span>+<span style='mso-tab-count:1'>     </span>Fibo13<span style='mso-tab-count:1'>      </span>= Output Lucas18<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:"Courier New";color:green'>Input (Fibo34<span class=GramE>)<span style='mso-spacerun:yes'>  </span>=</span> Fibo8<span style='mso-tab-count: 2'>       </span>+<span style='mso-tab-count:1'>     </span>Fibo21<span style='mso-tab-count:1'>      </span>= Output Lucas29<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:"Courier New";color:green'>Input (Fibo55<span class=GramE>)<span style='mso-spacerun:yes'>  </span>=</span> Fibo13<span style='mso-tab-count: 1'>      </span>+<span style='mso-tab-count:1'>     </span>Fibo34<span style='mso-tab-count:1'>      </span>= Output Lucas47<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:"Courier New";color:green'>Input (Fibo89<span class=GramE>)<span style='mso-spacerun:yes'>  </span>=</span> Fibo21<span style='mso-tab-count: 1'>      </span>+<span style='mso-tab-count:1'>     </span>Fibo55<span style='mso-tab-count:1'>      </span>= Output Lucas76<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:"Courier New";color:green'>Input (Fibo144) = Fibo34<span style='mso-tab-count:1'>      </span>+<span style='mso-tab-count:1'>     </span>Fibo89<span style='mso-tab-count:1'>      </span>= Output Lucas123<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><span style='font-size:10.0pt;font-family:Verdana;color:green'><o:p>&nbsp;</o:p></span></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;color:green'>& <span class=GramE>the</span> Input Fibonacci pattern continues with the sum of the previous two& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;color:green'>As well as both inner Fibonacci summation patterns which are just shifted 1 &amp; 3 respectively to create the Lucas Pattern.<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:red'>Current Analyzed Spiral &amp; Concentric Convergence Output Dual Integer Sequences& :<o:p></o:p></span></b></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p>&nbsp;</o:p></span></b></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Converging SQRT2 [1.41& ]: [** Spiral technique matches OEIS Output]<o:p></o:p></span></u></b></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:9.0pt; font-family:Verdana'>In comparison to the {OEIS} <span style='mso-tab-count: 1'> </span><span style='mso-spacerun:yes'> </span><u><span style='color:blue'><a href="http://www.research.att.com/~njas/sequences/A002965">http://www.research.att.com/~njas/sequences/A002965 </a><o:p></o:p></span></u></span></p> <p class=MsoNormal style='margin-left:2.0in;text-indent:.5in'><u><span style='font-size:9.0pt;font-family:Verdana;color:blue'><span style='mso-spacerun:yes'> </span><a href="http://www.research.att.com/~njas/sequences/A082766">http://www.research.att.com/~njas/sequences/A082766 </a></span></u><span style='font-size:9.0pt;font-family:Verdana'><o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Verdana'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New"'>OEIS {OEIS almost matches the above &quot;spiral technique&quot; except for the generating function.}</span></u></b><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><br> Natural Pattern of <span class=SpellE>Numer</span>: 1, 1, 2, 3, 4, 7, <span style='color:red'>10</span>, 17, 24, 41, <span style='color:red'>58</span>, 99, 140, 239, <span style='color:red'>338</span>, 577, 816, 1393, <span style='color:red'>1970</span>, 3363, 4756, 8119<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>Natural Pattern of <span class=SpellE>Denom</span>: 0, 1, 1, 1, 2, 3, 5, <span style='color:red'>7</span>, 12, 17, 29, <span style='color:red'>41</span>, 70, 99, 169, <span style='color:red'>239</span>, 408, 577, 985, <span style='color:red'>1393</span>, 2378, 3363, 5741, 8119,<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;color:red'>** <span class=GramE>red</span> is what we remove in  forced or  mirrored limits labeled  concentric convergence.<o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>2DCode SQRT2 Dual integer sequence output& :<o:p></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>1.) Lower Linear<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>Natural Pattern of <span class=SpellE>Numer</span>:<span style='mso-spacerun:yes'>  </span>1,4,7,24,41,140,239,816,1393,4756,8119<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>Natural Pattern of <span class=SpellE>Denom</span>: 1,3,5,17,29,99,169,577,985,3363,5741<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>2.) Upper Linear<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>Natural Pattern of <span class=SpellE>Numer</span>:<span style='mso-spacerun:yes'>  </span>2<span class=GramE>,3,10,17,58,99,338,577,1970,3363</span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>Natural Pattern of <span class=SpellE>Denom</span>: 1<span class=GramE>,2,7,12,41,70,239,408,1393,2378</span><o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana'>And of course, those two linear patterns merge to combine and create the documented OEIS pattern labeled by the writer as spiral convergence;<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=GramE><span style='font-size:10.0pt;font-family:Verdana'>given</span></span><span style='font-size:10.0pt;font-family:Verdana'> the limits are independent and sometimes the deviation is not necessarily closer then previous deviation. <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana'>Only, when isolated to the upper or the lower pattern of values that it actually fits the definition of convergence.<o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:blue'>3.)</span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'> </span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:blue'><a href="http://www.2dcode-r-past.com/Geometry/SQ/sqrt2_independent_converge.htm">Rotating convergence  [independent limits (B)] through 10K<span class=GramE>.</span></a></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'>. Lower Limit starts @ ZERO and Upper Limit = 2<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Numer</span>: 2,3,4,7,10,17,24,41,58,99,140,239,338,577,816,1393,1970,3363,4756,8119& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Denom</span>: 1,2,3,5,7,12,17,29,41,70,99,169,239,408,577,985,1393,2378,3363,5741<o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'><span style='mso-tab-count: 1'>          </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana'>Alternatively, if we take away&nbsp;that independent comparison and want to ensure that we actually converge closer each fractional approximation& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana'>Then we copy or mirror the same deviation to the opposite limit each hit. Hence, the label  forced convergence.<o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:red'>4.) <a href="http://www.2dcode-r-past.com/Geometry/SQ/sqrt2_mirrored_converge.htm"><span style='mso-bidi-font-family:"Times New Roman";color:windowtext;font-weight: normal;text-decoration:none;text-underline:none'><span style='mso-spacerun:yes'> </span></span><span style='color:red'>Forced convergence through  [mirrored limits (A)] through 10K</span></a></span></b><span class=GramE><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>..</span></b></span><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:"Courier New";color:black'> <o:p></o:p></span></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Concentric&nbsp;or Mirrored&nbsp;Pattern of <span class=SpellE>Numer</span>: 1,2,3,4,7,17,24,41,99,140,239,577,816,1393,3363,4756,8119& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Concentric or Mirrored&nbsp;Pattern of <span class=SpellE>Denom</span>: 1,1,2,3,5,12,17,29,70,99,169,408,577,985,2378,3363,5741<o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'><span style='mso-tab-count: 1'>          </span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>Once the pattern kicks in for SQRT2; it breaks down in groups of 3. <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>The limits control the first few fractional approximations; <span class=GramE>We</span> confirm the pattern on the 4th iteration. </span><o:p></o:p></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <table class=MsoNormalTable border=1 cellspacing=0 cellpadding=0 width=876 style='width:657.35pt;margin-left:35.25pt;border-collapse:collapse;border: none;mso-border-alt:solid windowtext .5pt;mso-padding-alt:0in 5.4pt 0in 5.4pt; mso-border-insideh:.5pt solid windowtext;mso-border-insidev:.5pt solid windowtext'> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;height:12.75pt'> <td width=48 nowrap valign=bottom style='width:36.35pt;border:solid windowtext 1.0pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>1<o:p></o:p></span></p> </td> <td width=408 valign=bottom style='width:4.25in;border:solid windowtext 1.0pt; border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt: solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> <td width=48 valign=bottom style='width:.5in;border:solid windowtext 1.0pt; border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt: solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>1<o:p></o:p></span></p> </td> <td width=372 nowrap valign=bottom style='width:279.0pt;border:solid windowtext 1.0pt; border-left:none;mso-border-left-alt:solid windowtext .5pt;mso-border-alt: solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>**1/1 Forced due to rollup analysis**<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:1;height:12.75pt'> <td width=48 nowrap valign=bottom style='width:36.35pt;border:solid windowtext 1.0pt; border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext .5pt; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>2<o:p></o:p></span></p> </td> <td width=408 valign=bottom style='width:4.25in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> <td width=48 valign=bottom style='width:.5in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>1<o:p></o:p></span></p> </td> <td width=372 nowrap valign=bottom style='width:279.0pt;border-top:none; border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:2;height:12.75pt'> <td width=48 nowrap valign=bottom style='width:36.35pt;border:solid windowtext 1.0pt; border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext .5pt; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>3<o:p></o:p></span></p> </td> <td width=408 valign=bottom style='width:4.25in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> <td width=48 valign=bottom style='width:.5in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>2<o:p></o:p></span></p> </td> <td width=372 nowrap valign=bottom style='width:279.0pt;border-top:none; border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:3;height:12.75pt'> <td width=48 nowrap valign=bottom style='width:36.35pt;border:solid windowtext 1.0pt; border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext .5pt; background:#FF99CC;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>4<o:p></o:p></span></p> </td> <td width=408 valign=bottom style='width:4.25in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> <td width=48 valign=bottom style='width:.5in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#FFFF99;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>3<o:p></o:p></span></p> </td> <td width=372 nowrap valign=bottom style='width:279.0pt;border-top:none; border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>**Begin pattern confirmation**<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:4;height:12.75pt'> <td width=48 nowrap valign=bottom style='width:36.35pt;border:solid windowtext 1.0pt; border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext .5pt; background:#FF99CC;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>7<o:p></o:p></span></p> </td> <td width=408 valign=bottom style='width:4.25in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> <td width=48 valign=bottom style='width:.5in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#FFFF99;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>5<o:p></o:p></span></p> </td> <td width=372 nowrap valign=bottom style='width:279.0pt;border-top:none; border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:5;height:12.75pt'> <td width=48 nowrap valign=bottom style='width:36.35pt;border:solid windowtext 1.0pt; border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext .5pt; background:#F3F3F3;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial;color:blue'>17<o:p></o:p></span></p> </td> <td width=408 valign=bottom style='width:4.25in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#F3F3F3;padding:0in 5.4pt 0in 5.4pt; height:12.75pt' u1:num> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial; color:blue'>=1+2+3+4+7 [rollup]<o:p></o:p></span></p> </td> <td width=48 valign=bottom style='width:.5in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#F3F3F3;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial;color:blue'>12<o:p></o:p></span></p> </td> <td width=372 nowrap valign=bottom style='width:279.0pt;border-top:none; border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#F3F3F3;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial; color:blue'>=1+1+2+3+5 [rollup]<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:6;height:12.75pt'> <td width=48 nowrap valign=bottom style='width:36.35pt;border:solid windowtext 1.0pt; border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext .5pt; background:#FF99CC;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>24<o:p></o:p></span></p> </td> <td width=408 valign=bottom style='width:4.25in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> <td width=48 valign=bottom style='width:.5in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#FFFF99;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>17<o:p></o:p></span></p> </td> <td width=372 nowrap valign=bottom style='width:279.0pt;border-top:none; border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:7;height:12.75pt'> <td width=48 nowrap valign=bottom style='width:36.35pt;border:solid windowtext 1.0pt; border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext .5pt; background:#FF99CC;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>41<o:p></o:p></span></p> </td> <td width=408 valign=bottom style='width:4.25in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> <td width=48 valign=bottom style='width:.5in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#FFFF99;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>29<o:p></o:p></span></p> </td> <td width=372 nowrap valign=bottom style='width:279.0pt;border-top:none; border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:8;height:12.75pt'> <td width=48 nowrap valign=bottom style='width:36.35pt;border:solid windowtext 1.0pt; border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext .5pt; background:#F3F3F3;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial;color:blue'>99<o:p></o:p></span></p> </td> <td width=408 valign=bottom style='width:4.25in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#F3F3F3;padding:0in 5.4pt 0in 5.4pt; height:12.75pt' u1:num> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial; color:blue'>=1+2+3+4+7+17+24+41 [rollup]<o:p></o:p></span></p> </td> <td width=48 valign=bottom style='width:.5in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#F3F3F3;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial;color:blue'>70<o:p></o:p></span></p> </td> <td width=372 nowrap valign=bottom style='width:279.0pt;border-top:none; border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#F3F3F3;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial; color:blue'>=1+1+2+3+5+12+17+29 [rollup]<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:9;height:12.75pt'> <td width=48 nowrap valign=bottom style='width:36.35pt;border:solid windowtext 1.0pt; border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext .5pt; background:#FF99CC;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>140<o:p></o:p></span></p> </td> <td width=408 valign=bottom style='width:4.25in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> <td width=48 valign=bottom style='width:.5in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#FFFF99;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>99<o:p></o:p></span></p> </td> <td width=372 nowrap valign=bottom style='width:279.0pt;border-top:none; border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:10;height:12.75pt'> <td width=48 nowrap valign=bottom style='width:36.35pt;border:solid windowtext 1.0pt; border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext .5pt; background:#FF99CC;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>239<o:p></o:p></span></p> </td> <td width=408 valign=bottom style='width:4.25in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> <td width=48 valign=bottom style='width:.5in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#FFFF99;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>169<o:p></o:p></span></p> </td> <td width=372 nowrap valign=bottom style='width:279.0pt;border-top:none; border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:11;height:12.75pt'> <td width=48 nowrap valign=bottom style='width:36.35pt;border:solid windowtext 1.0pt; border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext .5pt; background:#F3F3F3;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial;color:blue'>577<o:p></o:p></span></p> </td> <td width=408 valign=bottom style='width:4.25in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#F3F3F3;padding:0in 5.4pt 0in 5.4pt; height:12.75pt' u1:num> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial; color:blue'>=1+2+3+4+7+17+24+41+99+140+239 [rollup]<o:p></o:p></span></p> </td> <td width=48 valign=bottom style='width:.5in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#F3F3F3;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial;color:blue'>408<o:p></o:p></span></p> </td> <td width=372 nowrap valign=bottom style='width:279.0pt;border-top:none; border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#F3F3F3;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial; color:blue'>=1+1+2+3+5+12+17+29+70+99+169 [rollup]<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:12;height:12.75pt'> <td width=48 nowrap valign=bottom style='width:36.35pt;border:solid windowtext 1.0pt; border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext .5pt; background:#FF99CC;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>816<o:p></o:p></span></p> </td> <td width=408 valign=bottom style='width:4.25in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> <td width=48 valign=bottom style='width:.5in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#FFFF99;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>577<o:p></o:p></span></p> </td> <td width=372 nowrap valign=bottom style='width:279.0pt;border-top:none; border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:13;height:12.75pt'> <td width=48 nowrap valign=bottom style='width:36.35pt;border:solid windowtext 1.0pt; border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext .5pt; background:#FF99CC;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>1393<o:p></o:p></span></p> </td> <td width=408 valign=bottom style='width:4.25in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> <td width=48 valign=bottom style='width:.5in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#FFFF99;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>985<o:p></o:p></span></p> </td> <td width=372 nowrap valign=bottom style='width:279.0pt;border-top:none; border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:14;height:12.75pt'> <td width=48 nowrap valign=bottom style='width:36.35pt;border:solid windowtext 1.0pt; border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext .5pt; background:#F3F3F3;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial;color:blue'>3363<o:p></o:p></span></p> </td> <td width=408 valign=bottom style='width:4.25in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#F3F3F3;padding:0in 5.4pt 0in 5.4pt; height:12.75pt' u1:num> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial; color:blue'>=1+2+3+4+7+17+24+41+99+140+239+577+816+1393 [rollup]<o:p></o:p></span></p> </td> <td width=48 valign=bottom style='width:.5in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#F3F3F3;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial;color:blue'>2378<o:p></o:p></span></p> </td> <td width=372 nowrap valign=bottom style='width:279.0pt;border-top:none; border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#F3F3F3;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial; color:blue'>=1+1+2+3+5+12+17+29+70+99+169+408+577+985 [rollup]<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:15;height:12.75pt'> <td width=48 nowrap valign=bottom style='width:36.35pt;border:solid windowtext 1.0pt; border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext .5pt; background:#FF99CC;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>4756<o:p></o:p></span></p> </td> <td width=408 valign=bottom style='width:4.25in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> <td width=48 valign=bottom style='width:.5in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#FFFF99;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>3363<o:p></o:p></span></p> </td> <td width=372 nowrap valign=bottom style='width:279.0pt;border-top:none; border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:16;mso-yfti-lastrow:yes;height:12.75pt'> <td width=48 nowrap valign=bottom style='width:36.35pt;border:solid windowtext 1.0pt; border-top:none;mso-border-top-alt:solid windowtext .5pt;mso-border-alt:solid windowtext .5pt; background:#FF99CC;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>8119<o:p></o:p></span></p> </td> <td width=408 valign=bottom style='width:4.25in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt' u1:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> <td width=48 valign=bottom style='width:.5in;border-top:none;border-left: none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;background:#FFFF99;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>5741<o:p></o:p></span></p> </td> <td width=372 nowrap valign=bottom style='width:279.0pt;border-top:none; border-left:none;border-bottom:solid windowtext 1.0pt;border-right:solid windowtext 1.0pt; mso-border-top-alt:solid windowtext .5pt;mso-border-left-alt:solid windowtext .5pt; mso-border-alt:solid windowtext .5pt;padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> </tr> </table> <p class=MsoNormal style='margin-left:.5in'><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>** 1/1 Appears to be a requirement to the pattern forcing the upper and lower limits to conform to this  initial hit .<b style='mso-bidi-font-weight:normal'><o:p></o:p></b></span></i></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>Interestingly, this shows definitive alternating patterns of summation intermixing integer values similar; yet, more complex then how <span class=SpellE>fibo</span> works<span class=GramE>.{</span>via sum the last two}<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>There&nbsp;are numerous ways to define and relate these dual integer sequences; here is what appears to be a few of the simplest and or purest<span class=GramE>?:</span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New"'>Purple Numerator</span></u></b><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'> = sum the last two<span class=GramE>&nbsp; (</span>like <span class=SpellE>fibo</span>)<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><span style='mso-tab-count: 3'>                             </span>=<span class=SpellE>Numer</span>(x-1<span class=GramE>)+</span><span class=SpellE>Numer</span>(x-2) <b style='mso-bidi-font-weight: normal'>.OR.</b><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><span style='mso-tab-count: 2'>                   </span><span style='mso-tab-count:1'>          </span>=<span class=GramE>SUM(</span><span class=SpellE>Denom</span>(x-<span class=SpellE>nIteration</span>):<span class=SpellE>Denom</span>(x-1))<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:red'>Example:<o:p></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New"'>Numer</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>(</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>24)=<span class=SpellE>Numer</span>(7)+<span class=SpellE>Numer</span>(17) .OR.<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>Numer</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>(24)= <span class=SpellE>Denom</span>(1)+<span class=SpellE>Denom</span>(1)+<span class=SpellE>Denom</span>(2)+<span class=SpellE>Denom</span>(3)+<span class=SpellE>Denom</span>(5)+<span class=SpellE>Denom</span>(12)<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New"'>Numer</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>(</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>41)=<span class=SpellE>Numer</span>(17)+<span class=SpellE>Numer</span>(24) .OR.<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>Numer</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>(41)= <span class=SpellE>Denom</span>(1)+<span class=SpellE>Denom</span>(1)+<span class=SpellE>Denom</span>(2)+<span class=SpellE>Denom</span>(3)+<span class=SpellE>Denom</span>(5)+<span class=SpellE>Denom</span>(12)+<span class=SpellE>Denom</span>(17)<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New"'>Yellow Denominator</span></u></b><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'> = sum the last two (like <span class=SpellE>fibo</span>)<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><span style='mso-tab-count: 3'>                             </span>=<span class=SpellE>Denom</span>(x-1<span class=GramE>)+</span><span class=SpellE>Denom</span>(x-2) <b style='mso-bidi-font-weight: normal'>.OR.</b><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><span style='mso-tab-count: 3'>                             </span>=<span class=GramE>IIF(</span>MOD(<span class=SpellE>Denom</span>(x-1),2)=0,Numer(x-1),<span class=SpellE>Numer</span>(x-2)+<span class=SpellE>Denom</span>(x-2))<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:red'>Example:<o:p></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New"'>Denom</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>(</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>17)=<span class=SpellE>Denom</span>(5)+<span class=SpellE>Denom</span>(12) .OR.<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New"'>Denom</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>(</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>17)=<span class=SpellE>Numer</span>(17)<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New"'>Denom</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>(</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>29)=<span class=SpellE>Denom</span>(12)+<span class=SpellE>Denom</span>(17) .OR.<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New"'>Denom</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>(</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>29)=<span class=SpellE>Numer</span>(17)+<span class=SpellE>Denom</span>(12) [triangle]<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New"'>Any&nbsp;3rd {grey row} Denominator</span></u></b><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'> = sum last purple <span class=SpellE>numers+last</span> yellow <span class=SpellE>denom</span> .OR. <span class=GramE>a</span> summation rollup of all <span class=SpellE>denoms</span> up to {x-1} current position</span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><span style='mso-spacerun:yes'>  </span><span style='mso-tab-count:1'>        </span><span style='mso-spacerun:yes'>       </span><span style='mso-spacerun:yes'> </span><span style='mso-spacerun:yes'>   </span><span class=GramE>=(</span><span class=SpellE>Numer</span>(x-1)+ <span class=SpellE>Denom</span>(x-1)) .OR.</span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; =<span class=GramE>SUM(</span><span class=SpellE>Denom</span>(x-<span class=SpellE>nIteration</span>):<span class=SpellE>Denom</span>(x-1))<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>&nbsp;&nbsp;&nbsp; </span><span style='font-size:10.0pt;font-family:Arial'>&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp; </span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><strong><u><span style='font-size: 10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New";color:red'>Example:</span><span style='color:red'><o:p></o:p></span></u></strong></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New"'>Denom</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>(</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>12) = <span class=SpellE>Numer</span>(7)+<span class=SpellE>Denom</span>(5) <strong><span style='font-family:Verdana;mso-bidi-font-family:"Courier New"'>.OR.</span></strong></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>Denom</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>(12) = <span class=SpellE>Denom</span>(1)+<span class=SpellE>Denom</span>(1)+<span class=SpellE>Denom</span>(2)+<span class=SpellE>Denom</span>(3)+<span class=SpellE>Denom</span>(5)</span></p> <p class=MsoNormal style='margin-left:.5in'>&nbsp;</p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New"'>Denom</span></span></span><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>(</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>70) = <span class=SpellE>Numer</span>(41)+<span class=SpellE>Denom</span>(29) <strong><span style='font-family:Verdana;mso-bidi-font-family:"Courier New"'>.OR.</span></strong></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>Denom</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>(70) = <span class=SpellE>Denom</span>(1)+<span class=SpellE>Denom</span>(1)+<span class=SpellE>Denom</span>(2)+<span class=SpellE>Denom</span>(3)+<span class=SpellE>Denom</span>(5)+<span class=SpellE>Denom</span>(12)+<span class=SpellE>Denom</span>(17)+<span class=SpellE>Denom</span>(29)</span></p> <p class=MsoNormal style='margin-left:.5in'>&nbsp;</p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New"'>Any&nbsp;3rd {grey row} Numerator</span></u></b><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'> = summation group rollup of all <span class=SpellE>numers</span> leading to the grey <span class=SpellE>numer</span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><span style='mso-tab-count: 1'>          </span><span style='mso-spacerun:yes'>        </span>=<span class=GramE>SUM(</span><span class=SpellE>Numer</span>(x-<span class=SpellE>nIteration</span>):<span class=SpellE>Numer</span>(x-1))<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><strong><u><span style='font-size: 10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New";color:red'>Example:</span><span style='color:red'><o:p></o:p></span></u></strong></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>Numer</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>(17)=<span class=SpellE>Numer</span>(1)+ <span class=SpellE>Numer</span>(2)+ <span class=SpellE>Numer</span>(3)+ <span class=SpellE>Numer</span>(4)+ <span class=SpellE>Numer</span>(7)</span></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>Numer</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>(99)= <span class=SpellE>Numer</span>(1)+ <span class=SpellE>Numer</span>(2)+ <span class=SpellE>Numer</span>(3)+ <span class=SpellE>Numer</span>(4)+ <span class=SpellE>Numer</span>(7)+<span class=SpellE>Numer</span>(24)+<span class=SpellE>Numer</span>(41)<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>Consequently, we are summing up and converging closer. <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>Furthermore, these techniques appear to force one another; bonding both <span class=SpellE>numer</span> and <span class=SpellE>denom</span>&nbsp;to be exist in parallel. <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>Alternatively, we could easily view the grey summation values as the addition of the  <span style='color:red'>next OEIS or spiral value</span> (Above in red). </span><span style='font-size:10.0pt;font-family:Verdana'><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;color:red'>Missing Approximations</span><span style='font-size:10.0pt;font-family:Verdana'> bypassed via the forced or mirror technique as highlighted in red above:<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;color:red'>10/7<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;color:red'>58/41<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;color:red'>338/239<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;color:red'>1970/1393<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana'>Those missing approximations will sum with the previous values in relation to both grey <span class=SpellE>numer</span> and grey <span class=SpellE>denom</span> above. All 4 techniques interrelate.<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana'>The rest of the technique continues via simple summation& It still lacks an approved solid definition, but, the pattern is definitely there for analysis.<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana'>Furthermore, all&nbsp;4 techniques fit like puzzle pieces and are inter relational via more then just one function. </span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'><o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>The {grey rows} summation rollup pattern could relate to an approximation of 1 degree of Pi& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>Numer</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>:<span style='mso-spacerun:yes'>  </span>17, 99, 577, 3363& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span class=SpellE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>Denom</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>: 12, 70, 408, 2378& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>1 degree as it relates to Pi=<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>(4*<span class=SpellE><span class=GramE>Atan</span></span><span class=GramE>(</span>1))/180=</span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg"'>0.017453292519943295769236& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg"'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg"'>In comparison:<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg"'>(99/577)/10<span style='mso-spacerun:yes'>   </span>=0.01715771230502599653& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg"'>(577/3363)/10=0.017157300029735355337& <o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Converging SQRT3 [</span></u></b><b style='mso-bidi-font-weight:normal'><u><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg";color:black'>1.732050807568877</span></u></b><b style='mso-bidi-font-weight:normal'><u><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>& ]<o:p></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Lower Linear Numerator Pattern.....: 1,3,5,12,19,45,71,168,265,627,989,2340,3691,8733<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Lower Linear Denominator Pattern...: 1<span class=GramE>,2,3</span>, 7,11,26,41, 97,153,362,571,1351,2131,5042<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Upper Linear Numerator Pattern.....: 2<span class=GramE>,7,26,97,362,1351,5042</span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Upper Linear Denominator Pattern...: 1<span class=GramE>,4,15,56,209</span>, 780,2911<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana'>Again, those two linear patterns merge to combine and create the documented OEIS pattern.<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Spiral Numerator Pattern...........: 1,2,3,5,7,12,19,26,45,71,97,168,265,362,627,989,1351,2340,3691,5042,8733<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Spiral Denominator Pattern.........: 1,1,2,3,4, 7,11,15,26,41,56, 97,153,209,362,571, 780,1351,2131,2911,5042<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Unlike SQRT2, The SQRT3 mirrored limit or forced convergence technique is exactly the same output as the independent convergence output. <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Consequently, this forces many more data sets prior to any conclusive patterns can be deduced across all input in relation to the output.<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Concentric Numerator Pattern.......: 1,2,3,5,7,12,19,26,45,71,97,168,265,362,627,989,1351,2340,3691,5042,8733<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Concentric Denominator Pattern.....: 1,1,2,3,4,7,11,15,26,41,56,97,153,209,362,571,780,1351,2131,2911,5042<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:red'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:blue'>Raw SQRT3 fractional iteration data for analysis {both mirror and independent are the same}:<o:p></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:blue'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:red'>A.) <a href="http://www.2dcode-r-past.com/Geometry/SQ/sqrt3_upper_or_lower.htm"><span style='mso-bidi-font-family:"Times New Roman";color:windowtext;text-decoration: none;text-underline:none'><span style='mso-spacerun:yes'> </span></span><span style='color:red'>Forced and or Independent convergence {both the same output<span class=GramE>}<span style='mso-spacerun:yes'>  </span>through</span><span style='mso-spacerun:yes'>   </span><span class=SpellE>through</span> 10K</span></a><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:.5in'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'>[Patterns in Numerator/Denominator =ROOT3 <span class=SpellE><span class=GramE>apprx</span></span><span class=GramE> ]</span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><u><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>B.) ADDITIONAL SQRT3 DATA BREAK DOWN: <u><a href="http://www.2dcode-r-past.com/Geometry/SQ/SQRT3_Pattern.htm"><span style='color:black'>http://www.2dcode-r-past.com/Geometry/SQ/SQRT3_Pattern.htm</span></a> <o:p></o:p></u></span></p> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Converging <span class=GramE>SQRT(</span>3)/SQRT(2)<span style='mso-spacerun:yes'>  </span>Also known as SQRT(1.5)</span></u></b><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:"Courier New";color:black'> </span></b><b style='mso-bidi-font-weight:normal'><u><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:blue'>[<span class=SpellE>Wiki</span>:<span style='mso-spacerun:yes'>  </span><a href="http://en.wikipedia.org/wiki/Vesica_piscis">key <span class=SpellE>Vesica</span> Pisces</a> <span class=SpellE>Contsant</span>]</span></u></b><b style='mso-bidi-font-weight:normal'><u><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'> <o:p></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>A.) </span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:blue'><a href="http://www.2dcode-r-past.com/Geometry/SQ/vesica_mirrored_converge.htm"><span style='color:red'>Forced convergence through  [mirrored limits (A)] through 10K</span><span class=GramE>.</span></a></span><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>. Lower Limit starts @ ZERO and Upper Limit = 1.5<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Numer</span>: 3,4,5,6,11,38,49,60,109,376,485,594,1079,3722,4801,5880& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Denom</span>: 2,3,4,5,9,31,40,49,89,307,396,485,881,3039,3920,4801<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>B.) </span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:blue'><a href="http://www.2dcode-r-past.com/Geometry/SQ/vesica_independent_converge.htm">Rotating convergence  [independent limits (B)] through 10K.</a></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'>. Lower Limit starts @ ZERO and Upper Limit = 1.5<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Numer</span>: 2,3,4,5,6,11,16,27,38,49,60,109,158,267,376,485,594& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Denom</span>: 1,2,3,4,5,9,13,22,31,40,49,89,129,218,307,396,485<o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'><span style='mso-tab-count: 1'>          </span><a href="http://www.research.att.com/~njas/sequences/A142239"><span style='color:black'>Listed @ ATT Integer Sequence Encyclopedia</span></a><o:p></o:p></span></p> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'><span style='mso-spacerun:yes'> </span><o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Converging Pi [3.14& <span class=GramE>] :</span><o:p></o:p></span></u></b></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:red'>1.) <a href="http://www.2dcode-r-past.com/Geometry/SQ/pi_converge_upper_lower.htm"><span style='mso-bidi-font-family:"Times New Roman";color:windowtext;font-weight: normal;text-decoration:none;text-underline:none'><span style='mso-spacerun:yes'> </span></span><span style='color:red'>Forced convergence through  [mirrored limits <span style='mso-bookmark:OLE_LINK3'><span style='mso-bookmark:OLE_LINK2'>(A)</span></span>] through 100K</span></a><![if !supportNestedAnchors]><a name="OLE_LINK2"></a><a name="OLE_LINK3"></a><![endif]></span></b><span class=GramE><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>..</span></b></span><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:"Courier New";color:black'> <span class=GramE>upper</span> or lower {closer}& </span></b><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'> depending on the deviation of current Pi hit. [Patterns in Numerator/Denominator =<span class=GramE>Pi ]</span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:blue'>2.)<b style='mso-bidi-font-weight:normal'> <a href="http://www.2dcode-r-past.com/Geometry/SQ/pi_converge_upper_or_lower.htm">Rotating convergence  [independent limits (B)] through 10K</a><span class=GramE>..</span></b></span><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:"Courier New";color:black'> <span class=GramE>upper</span> then lower& </span></b><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'> depending on the deviation of current Pi hit. [Patterns in Numerator/Denominator =<span class=GramE>Pi ]</span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><a href="http://www.grahamhancock.com/phorum/read.php?f=1&amp;i=249980&amp;t=249505"><span style='color:black'>Posted @ GHMB</span></a><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><span style='mso-spacerun:yes'> </span><o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Converging Phi [1.618& ]:<o:p></o:p></span></u></b></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:red'>1.) <a href="http://www.2dcode-r-past.com/Geometry/SQ/Phi1_618.htm"><span style='mso-bidi-font-family:"Times New Roman";color:windowtext;text-decoration: none;text-underline:none'><span style='mso-spacerun:yes'> </span></span><span style='color:red'>Forced convergence through  [mirrored limits (A)] through 10K</span></a><span class=GramE>.<span style='color:black'>.</span></span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'> <span class=GramE>upper</span> then lower& depending on the deviation of current Phi<span style='mso-spacerun:yes'>  </span>hit. [Patterns in Numerator/Denominator =<span class=GramE>Phi ]</span><o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Converging phi [.618& ]:<o:p></o:p></span></u></b></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:red'>1.) <a href="http://www.2dcode-r-past.com/Geometry/SQ/Phi_618.htm"><span style='mso-bidi-font-family:"Times New Roman";color:windowtext;text-decoration: none;text-underline:none'><span style='mso-spacerun:yes'> </span></span><span style='color:red'>Forced convergence through  [mirrored limits (A)] through 10K</span></a></span><span class=GramE><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>..</span></span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'> <span class=GramE>upper</span> then lower& depending on the deviation of current phi<span style='mso-spacerun:yes'>  </span>hit. [Patterns in Numerator/Denominator =<span class=GramE>phi ]</span><o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='text-indent:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Converging SQRT5 [2.23& ]<o:p></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>SQRT5 Upper Linear Numerators...: 3,5,7,9,47,85,123,161,843,1525,2207,2889<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>SQRT5 Upper Linear <span class=GramE>Denominators.:</span> 1,2,3,4,21,38,55,72,377,682,987,1292<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>SQRT5 Lower Linear Numerators...: 1,2,11,20,29,38,199,360,521,682,3571,6460,9349<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>SQRT5 Lower Linear <span class=GramE>Denominators.:</span> 1,1,5,9,13,17,89,161,233,305,1597,2889,4181<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>SQRT5 Spiral Numerators.........: 1,2,3,5,7,9,11,20,29,38,47,85,123,161,199,360,521,682,843,1525,2207,2889,3571,6460,9349<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>SQRT5 Spiral<span style='mso-spacerun:yes'>  </span>Denominators......: 1,1,1,2,3,4,5,9,13,17,21,38,55,72,89,161,233,305,377,682,987,1292,1597,2889,4181<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>SQRT5 Concentric Numerators.....: 1,2,7,9,20,29,38,123,161,360,521,682,2207,2889,6460,9349<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>SQRT5 Concentric Denominators...: 1,1,3,4,9,13,17,55,72,161,233,305,987,1292,2889,4181<o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:red'>1.) <b style='mso-bidi-font-weight:normal'><a href="http://www.2dcode-r-past.com/Geometry/SQ/sqrt5_upper_or_lower.htm"><span style='mso-bidi-font-family:"Times New Roman";color:windowtext;font-weight: normal;text-decoration:none;text-underline:none'><span style='mso-spacerun:yes'> </span></span><span style='color:red'>Forced convergence through  [mirrored limits (A)] through 10K</span></a></b></span><span class=GramE><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>..</span></b></span><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:"Courier New";color:black'> <span class=GramE>upper</span> then lower</span></b><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'>& depending on the deviation of current SQRT5 <span class=SpellE>apprx</span> hit. [Patterns in Numerator/Denominator =ROOT5 <span class=SpellE><span class=GramE>apprx</span></span><span class=GramE> ]</span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:1.0in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <table class=MsoNormalTable border=0 cellspacing=0 cellpadding=0 width=331 style='width:248.0pt;margin-left:35.25pt;border-collapse:collapse;mso-padding-alt: 0in 5.4pt 0in 5.4pt'> <tr style='mso-yfti-irow:0;mso-yfti-firstrow:yes;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>1<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;padding:0in 5.4pt 0in 5.4pt; height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>1<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:1;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;padding:0in 5.4pt 0in 5.4pt; height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>2<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;padding:0in 5.4pt 0in 5.4pt; height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>1<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:2;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;padding:0in 5.4pt 0in 5.4pt; height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>7<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;padding:0in 5.4pt 0in 5.4pt; height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>3<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:3;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:#FFFF99; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>9<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;background:#FFCC99; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>4<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:4;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:#FFCC99; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>20<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;background:#FFFF99; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>9<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:5;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:#CCFFCC; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>29<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;background:#CCFFCC; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>13<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:6;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:lime; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>38<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;background:lime; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>17<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:7;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:#00CCFF; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>123<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;background:#FF99CC; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial;color:black'>55<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:8;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:#FFFF99; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>161<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;background:#FFCC99; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>72<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:9;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:#FFCC99; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>360<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;background:#FFFF99; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>161<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:10;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:#CCFFCC; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>521<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;background:#CCFFCC; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>233<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:11;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:lime; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>682<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;background:lime; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>305<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:12;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:#00CCFF; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>2207<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;background:#FF99CC; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial;color:black'>987<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:13;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:#FFFF99; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>2889<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;background:#FFCC99; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>1292<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:14;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:#FFCC99; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>6460<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;background:#FFFF99; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>2889<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:15;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:#CCFFCC; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>9349<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;background:#CCFFCC; padding:0in 5.4pt 0in 5.4pt;height:12.75pt' x:num> <p class=MsoNormal align=right style='text-align:right'><span style='font-size:10.0pt;font-family:Arial'>4181<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:16;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:17;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial'><o:p>&nbsp;</o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:18;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:#FFFF99; padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial'>&nbsp;<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial'>sum the last two like <span class=SpellE>fibo</span><o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:19;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:#FFCC99; padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial'>&nbsp;<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial'>5*<span class=SpellE>Denom</span>(x-1)<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:20;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:#00CCFF; padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial'>&nbsp;<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal><span class=SpellE><span style='font-size:10.0pt; font-family:Arial'>Numer</span></span><span style='font-size:10.0pt; font-family:Arial'>(x-1)+(5*<span class=SpellE>Denom</span>(x-1))<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:21;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:#CCFFCC; padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial'>&nbsp;<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial'>sum the last two like <span class=SpellE>fibo</span><o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:22;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:#FF99CC; padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial'>&nbsp;<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal><span class=SpellE><span style='font-size:10.0pt; font-family:Arial'>Numer</span></span><span style='font-size:10.0pt; font-family:Arial'>(x-1)+<span class=SpellE>Denom</span>(x-1)<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:23;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:lime; padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial'>&nbsp;<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal><span class=SpellE><span style='font-size:10.0pt; font-family:Arial'>Numer</span></span><span style='font-size:10.0pt; font-family:Arial'>(x-1)+<span class=SpellE>Numer</span>(x-3)<o:p></o:p></span></p> </td> </tr> <tr style='mso-yfti-irow:24;mso-yfti-lastrow:yes;height:12.75pt'> <td width=141 nowrap valign=bottom style='width:106.0pt;background:lime; padding:0in 5.4pt 0in 5.4pt;height:12.75pt'> <p class=MsoNormal><span style='font-size:10.0pt;font-family:Arial'>&nbsp;<o:p></o:p></span></p> </td> <td width=189 nowrap valign=bottom style='width:142.0pt;padding:0in 5.4pt 0in 5.4pt; height:12.75pt'> <p class=MsoNormal><span class=SpellE><span style='font-size:10.0pt; font-family:Arial'>Denom</span></span><span style='font-size:10.0pt; font-family:Arial'>(x-1)+<span class=SpellE>Denom</span>(x-3)<o:p></o:p></span></p> </td> </tr> </table> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>** We can also see SQRT5 <a href="http://www.2dcode-r-past.com/Geometry/SQ/Phi1_618.htm"><span style='color:black'>converge through the combination of Phi and phi</span></a> Golden string convergence pattern which shows that by combining two fractions will provide another <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Method of convergence combining the reciprocal & <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Converging SQRT6 [2.449& ]<o:p></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:blue'>1) <a href="http://www.2dcode-r-past.com/Geometry/SQ/sqrt6.htm">Rotating convergence  [independent limits (B)] through 999</a></span><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'>..</span></span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'> <span class=GramE>upper</span> then lower& depending on the deviation of current SQRT6 <span class=SpellE>apprx</span> hit. [Patterns in Numerator/Denominator =ROOT6 <span class=SpellE><span class=GramE>apprx</span></span><span class=GramE> ]</span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Numer</span>: 5,22,27,49,71,120,169,218,267,485,703& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Denom</span>: 2,9,11,20,29,49,69,89,109,198,287& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Converging SQRT7 [2.645& ]<o:p></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:blue'>1) <a href="http://www.2dcode-r-past.com/Geometry/SQ/sqrt7.htm">Rotating convergence  [independent limits (B)] through 999</a></span><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'>..</span></span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'> <span class=GramE>upper</span> then lower& depending on the deviation of current SQRT7 <span class=SpellE>apprx</span> hit. [Patterns in Numerator/Denominator =ROOT7 <span class=SpellE><span class=GramE>apprx</span></span><span class=GramE> ]</span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Numer</span>: 8,13,21,29,37,45,82,127,209,336,463,590& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Denom</span>: 3,5,8,11,14,17,31,48,79,127,175,223& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><a name="OLE_LINK8"></a><a name="OLE_LINK7"><span style='mso-bookmark:OLE_LINK8'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Converging SQRT8 [2.828& ]<o:p></o:p></span></u></b></span></a></p> <p class=MsoNormal style='margin-left:.5in'><span style='mso-bookmark:OLE_LINK7'><span style='mso-bookmark:OLE_LINK8'><b style='mso-bidi-font-weight:normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='mso-bookmark:OLE_LINK7'><span style='mso-bookmark:OLE_LINK8'><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:blue'>1) </span></span></span><a href="http://www.2dcode-r-past.com/Geometry/SQ/sqrt8.htm"><span style='mso-bookmark:OLE_LINK7'><span style='mso-bookmark:OLE_LINK8'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>Rotating convergence  [independent limits (B)] through 999</span></span></span><span style='mso-bookmark:OLE_LINK7'><span style='mso-bookmark:OLE_LINK8'></span></span></a><span style='mso-bookmark:OLE_LINK7'><span style='mso-bookmark:OLE_LINK8'><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>..</span></span></span></span><span style='mso-bookmark:OLE_LINK7'><span style='mso-bookmark:OLE_LINK8'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'> <span class=GramE>upper</span> then lower& depending on the deviation of current SQRT8 <span class=SpellE>apprx</span> hit. [Patterns in Numerator/Denominator =ROOT8 <span class=SpellE><span class=GramE>apprx</span></span><span class=GramE> ]</span><o:p></o:p></span></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='mso-bookmark:OLE_LINK7'><span style='mso-bookmark:OLE_LINK8'><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='mso-bookmark:OLE_LINK7'><span style='mso-bookmark:OLE_LINK8'><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Numer</span>: 14,17,31,48,65,82,<a name="OLE_LINK6"></a><a name="OLE_LINK5"><span style='mso-bookmark:OLE_LINK6'>99,181,280</span></a>,379,478,577& <o:p></o:p></span></span></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='mso-bookmark:OLE_LINK7'><span style='mso-bookmark:OLE_LINK8'><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Denom</span>: 5,6,11,17,23,29,35,64,99,134,169,204& <o:p></o:p></span></span></span></p> <span style='mso-bookmark:OLE_LINK8'></span><span style='mso-bookmark:OLE_LINK7'></span> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Converging SQRT10 [3.162277& ]<o:p></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:blue'>1) <a href="http://www.2dcode-r-past.com/Geometry/SQ/sqrt10.htm">Rotating convergence  [independent limits (B)] through 999</a></span><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'>..</span></span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'> <span class=GramE>upper</span> then lower& depending on the deviation of current SQRT10 <span class=SpellE>apprx</span> hit. [Patterns in Numerator/Denominator =ROOT10 <span class=SpellE><span class=GramE>apprx</span></span><span class=GramE> ]</span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Numer</span>: 19,22,41,60,79,98,117,136,253,370,487,604,721,838& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Denom</span>: 6,7,13,19,25,31,37,43,80,117,154,191,228,265& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Converging SQRT11 [3.3166& ]<o:p></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:blue'>1) <a href="http://www.2dcode-r-past.com/Geometry/SQ/sqrt11.htm">Rotating convergence  [independent limits (B)] through 999</a></span><span class=GramE><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'>..</span></span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'> <span class=GramE>upper</span> then lower& depending on the deviation of current SQRT11 <span class=SpellE>apprx</span> hit. [Patterns in Numerator/Denominator =ROOT11 <span class=SpellE><span class=GramE>apprx</span></span><span class=GramE> ]</span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Numer</span>: 7,10,13,23,33,43,53,63,73,136,199,262,461,660,859& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Denom</span>: 2,3,4,7,10,13,16,19,22,41,60,79,139,199,259& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Converging Chaos [4.6692 (Exact)<span class=GramE>]</span></span></u></b><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:"Courier New";color:red'><a href="http://en.wikipedia.org/wiki/Chaos"><span style='color:red'>[<span class=SpellE>Wiki</span>: Chaos]</span></a></span></b><b style='mso-bidi-font-weight: normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p></o:p></span></b></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:blue'>1) <a href="http://www.2dcode-r-past.com/Geometry/SQ/chaos.htm">Rotating convergence  [independent limits (B)] through 999</a>9</span><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'> [10K]<span class=GramE>..</span> <span class=GramE>upper</span> then lower& depending on the deviation of current Chaos <span class=SpellE>apprx</span> hit. [Patterns in Numerator/Denominator =Chaos <span class=SpellE><span class=GramE>apprx</span></span><span class=GramE> ]</span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Numer</span>: 14,467,481,495,509,523,537,551,565,579,593,607,621,1228,1849,3077,4305,5533,6761,7989,9217& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>Natural Pattern of <span class=SpellE>Denom</span>: 3,100,103,106,109,112,115,118,121,124,127,130,133,263,396,659,922,1185,1448,1711,1974& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'>Converging  E [2.71282<span class=GramE>]</span></span></u></b><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:"Courier New";color:red'><a href="http://en.wikipedia.org/wiki/E_(mathematical_constant)"><span style='color:red'>[<span class=SpellE>Wiki</span>: Natural Constant E]</span></a></span></b><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family: Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p></o:p></span></b></p> <p class=MsoNormal style='margin-left:.5in'><b style='mso-bidi-font-weight: normal'><u><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family: "Courier New";color:black'><o:p><span style='text-decoration:none'>&nbsp;</span></o:p></span></u></b></p> <p class=MsoNormal style='margin-left:.75in;text-indent:-.25in;mso-list:l1 level1 lfo4; tab-stops:list .75in'><![if !supportLists]><span style='font-size:10.0pt; font-family:Verdana;mso-fareast-font-family:Verdana;mso-bidi-font-family:Verdana; color:black'><span style='mso-list:Ignore'>1)<span style='font:7.0pt "Times New Roman"'>&nbsp;&nbsp;&nbsp;&nbsp; </span></span></span><![endif]><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:blue'><a href="http://www.2dcode-r-past.com/Geometry/SQ/elog.htm">Rotating convergence  [independent limits (B)] through 999</a> </span><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>[1K]<span class=GramE>..</span> <span class=GramE>upper</span> then lower& depending on the deviation of current  E log <span class=SpellE>apprx</span> hit. <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in;text-indent:.25in'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"; color:black'>[Patterns in Numerator/Denominator = E Log <span class=SpellE><span class=GramE>apprx</span></span><span class=GramE> ]</span><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:Arial'>Research &amp; Analysis of common cycles modeled around the BASE &quot;year&quot; in days; utilizing the same programmatic techniques applied on the constants above. </span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>- <a href="http://www.2dcode-r-past.com/Geometry/SQ/Convering_Cycles.htm">Converging cycles: [solar cycle, sidereal, Gregorian, <span class=SpellE>lunar<span class=GramE>..etc</span></span>]</a><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana'><o:p>&nbsp;</o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New";color:black'>A simple fraction in itself can contain a pattern<span class=GramE>..</span> <span class=GramE>take</span> 999/998 for example =<o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg";color:black'>1.00</span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg"; color:red'>1</span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"MS Shell Dlg";color:black'>00</span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg"; color:red'>2</span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"MS Shell Dlg";color:black'>00</span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg"; color:red'>4</span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"MS Shell Dlg";color:black'>00</span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg"; color:red'>8</span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"MS Shell Dlg";color:black'>0</span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg"; color:red'>16</span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"MS Shell Dlg";color:black'>0</span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg"; color:red'>32</span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"MS Shell Dlg";color:black'>0</span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg"; color:red'>64128</span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"MS Shell Dlg";color:blue'>256</span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg"; color:lime'>5</span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"MS Shell Dlg";color:green'>13</span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg"; color:black'>0</span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"MS Shell Dlg";color:green'>26</span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg"; color:black'>0</span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"MS Shell Dlg";color:green'>52104208416</span><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg"; color:red'>833</span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"MS Shell Dlg";color:black'>6673346693386773547094188<o:p></o:p></span></p> <p class=MsoNormal style='text-indent:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"MS Shell Dlg";color:black'>Interesting to note where 256 should double to 512& instead becomes 513 and then begins to double again& . </span><span style='font-size:10.0pt;font-family:Verdana; mso-bidi-font-family:"Courier New";color:black'><o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'>Also 416 should have doubled to 832, yet adds an extra  1 to end up at 833& <o:p></o:p></span></p> <p class=MsoNormal style='margin-left:.5in'><span style='font-size:10.0pt; font-family:Verdana;mso-bidi-font-family:"Courier New"'><a href="http://www.grahamhancock.com/phorum/read.php?f=1&amp;i=242648&amp;t=242648"></a></span><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none; text-underline:none'><o:p></o:p></span></span></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'><a href="http://www.grahamhancock.com/phorum/read.php?f=1&amp;i=242648&amp;t=242648"><span style='color:windowtext;text-decoration:none;text-underline:none'><span style='mso-tab-count:1'>            </span><span class=GramE>Thank<span style='mso-spacerun:yes'>  </span>you</span> for your review.</span></a><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none; text-underline:none'><o:p></o:p></span></span></span></b></p> <p class=MsoNormal><b style='mso-bidi-font-weight:normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'><a href="http://www.grahamhancock.com/phorum/read.php?f=1&amp;i=242648&amp;t=242648"><span style='color:windowtext;text-decoration:none;text-underline:none'><span style='mso-tab-count:1'>            </span>Rob Miller</span></a><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none; text-underline:none'><o:p></o:p></span></span></span></b></p> <p class=MsoNormal><o:p>&nbsp;</o:p></p> <p class=MsoNormal style='text-indent:.5in'><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'>This www is intended for personal and collaborative collection of research / data 4 analyses.<o:p></o:p></span></i></p> <p class=MsoNormal style='text-indent:.5in'><i style='mso-bidi-font-style:normal'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'><span style='mso-spacerun:yes'> </span>No determinations of fact have been made. 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Permission and access to all material will be granted on an individual basis.</span></a><span class=MsoHyperlink><span style='color:windowtext;text-decoration:none; text-underline:none'><o:p></o:p></span></span></span></p> <p class=MsoNormal style='tab-stops:45.8pt 91.6pt 137.4pt 183.2pt 229.0pt 274.8pt 320.6pt 366.4pt 412.2pt 458.0pt 503.8pt 549.6pt 595.4pt 641.2pt 687.0pt 732.8pt'><span style='font-size:10.0pt;font-family:Verdana;mso-bidi-font-family:"Courier New"'><a href="http://www.grahamhancock.com/phorum/read.php?f=1&amp;i=242648&amp;t=242648"><span style='color:windowtext;text-decoration:none;text-underline:none'><span style='mso-spacerun:yes'>               </span>(<b style='mso-bidi-font-weight: normal'>Copyright 2dCODE-R-PAST.com 2008<span class=GramE>-)</span></b></span></a><span class=MsoHyperlink><b style='mso-bidi-font-weight:normal'><span style='color:windowtext;text-decoration:none;text-underline:none'><o:p></o:p></span></b></span></span></p> </div> </body> </html>