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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 3.2 Final//EN"> <html> <head> <link rel="stylesheet" href="/styles.css"> <meta name="format-detection" content="telephone=no"> <meta http-equiv="content-type" content="text/html; charset=utf-8"> <meta name=viewport content="width=device-width, initial-scale=1"> <meta name="keywords" content="OEIS,integer sequences,Sloane" /> <title>The On-Line Encyclopedia of Integer Sequences (OEIS)</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/history?seq=A119812" function redir() { var host = document.location.hostname; if(host != "oeis.org" && host != "127.0.0.1" && !/^([0-9.]+)$/.test(host) && host != "localhost" && host != "localhost.localdomain") { document.location = "https"+":"+"//"+"oeis"+".org/" + myURL; } } function sf() { if(document.location.pathname == "/" && document.f) document.f.q.focus(); } </script> </head> <body bgcolor=#ffffff onload="redir();sf()"> <div class=loginbar> <div class=login> <a href="/login?redirect=%2fhistory%3fseq%3dA119812">login</a> </div> </div> <div class=center><div class=top> <center> <div class=donors> The OEIS is supported by <a href="http://oeisf.org/#DONATE">the many generous donors to the OEIS Foundation</a>. </div> <div class=banner> <a href="/"><img class=banner border="0" width="600" src="/banner2021.jpg" alt="The On-Line Encyclopedia of Integer Sequences (OEIS)"></a> </div> </center> </div></div> <div class=center><div class=pagebody> <div class=searchbarcenter> <form name=f action="/search" method="GET"> <div class=searchbargreet> <div class=searchbar> <div class=searchq> <input class=searchbox maxLength=1024 name=q value="" title="Search Query"> </div> <div class=searchsubmit> <input type=submit value="Search" name=go> </div> <div class=hints> <span class=hints><a href="/hints.html">Hints</a></span> </div> </div> <div class=searchgreet> (Greetings from <a href="/welcome">The On-Line Encyclopedia of Integer Sequences</a>!) </div> </div> </form> </div> <center> <h2 style="margin-top:0; margin-bottom:0;"> Revision History for <a href="/A119812">A119812</a> </h2> (Bold, blue-underlined text is an <ins>addition</ins>; faded, red-underlined text is a <del>deletion</del>.) </center> <br> Showing all changes. <br> <div class=history> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum><a href="/A119812">A119812</a></div> <div class=seqname> Decimal expansion of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} <a href="/A049472" title="a(n) = floor(n/sqrt(2)).">A049472</a>(n)/2^n = Sum_{n>=1} 1/2^<a href="/A001951" title="A Beatty sequence: a(n) = floor(n*sqrt(2)).">A001951</a>(n). <br> <font size=-1>(<a href="/history?seq=A119812">history</a>; <a href="/A119812">published version</a>) </font> </div> </div> </div> <div class=resultline></div> <div class="revbar"> <a href="/history/view?seq=A119812&v=9">#9</a> by <a href="/history?user=Bruno%20Berselli">Bruno Berselli</a> at Fri Nov 22 10:13:00 EST 2013 </div> <div> <div class=entry> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <p class="diffs"><tt><del>proposed</del></tt></p> <p class="diffs"><tt><ins>approved</ins></tt></p> </div> </div> </div> </div> <div class=space4></div> <div class=resultline></div> <div class="revbar"> <a href="/history/view?seq=A119812&v=8">#8</a> by <a href="/history?user=Peter%20Bala">Peter Bala</a> at Fri Nov 22 06:46:47 EST 2013 </div> <div> <div class=entry> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <p class="diffs"><tt><del>editing</del></tt></p> <p class="diffs"><tt><ins>proposed</ins></tt></p> </div> </div> </div> </div> <div class=space4></div> <div class=resultline></div> <div class="revbar"> <a href="/history/view?seq=A119812&v=7">#7</a> by <a href="/history?user=Peter%20Bala">Peter Bala</a> at Fri Nov 22 06:45:16 EST 2013 </div> <div> <div class=entry> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <p class="diffs"><tt><ins>W. W. Adams and J. L. Davison, <a href="http://www.jstor.org/stable/2041889">A remarkable class of continued fractions</a>, Proc. Amer. Math. Soc. 65 (1977), 194-198.</ins></tt></p> <p class="diffs"><tt><ins>P. G. Anderson, T. C. Brown, P. J.-S. Shiue, <a href="http://people.math.sfu.ca/~vjungic/tbrown/tom-28.pdf">A simple proof of a remarkable continued fraction identity</a> Proc. Amer. Math. Soc. 123 (1995), 2005-2009.</ins></tt></p> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <p class="diffs"><tt><del>approved</del></tt></p> <p class="diffs"><tt><ins>editing</ins></tt></p> </div> </div> </div> </div> <div class="space4"></div> <div class=discussbar> <div class=discusshead>Discussion</div> <div class=discussnote> <div class=date>Fri Nov 22</div> <div class=time>06:46</div> <pre class=note><span class=user>Peter Bala</span>: The links give a proof of the continued fraction expansion.</pre> </div> </div> <div class=space4></div> <div class=resultline></div> <div class="revbar"> <a href="/history/view?seq=A119812&v=6">#6</a> by <a href="/history?user=Russ%20Cox">Russ Cox</a> at Fri Mar 30 18:36:57 EDT 2012 </div> <div> <div class=entry> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <p class="diffs"><tt><ins>_</ins>Paul D. Hanna<del> </del><del>(</del><del>pauldhanna</del><del>(</del><del>AT</del><del>)</del><del>juno</del><del>.</del><del>com</del><del>)</del><del>,</del><del> </del><ins>_</ins><ins>,</ins><ins> </ins>May 26 2006</tt></p> </div> </div> </div> </div> <div class="space4"></div> <div class=discussbar> <div class=discusshead>Discussion</div> <div class=discussnote> <div class=date>Fri Mar 30</div> <div class=time>18:36</div> <pre class=note><span class=user>OEIS Server</span>: https://oeis.org/edit/global/213</pre> </div> </div> <div class=space4></div> <div class=resultline></div> <div class="revbar"> <a href="/history/view?seq=A119812&v=5">#5</a> by <a href="/history?user=Russ%20Cox">Russ Cox</a> at Fri Mar 30 17:39:12 EDT 2012 </div> <div> <div class=entry> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <p class="diffs"><tt>Removed leading zero and corrected offset <ins>_</ins>R. J. Mathar<del> </del><del>(</del><del>mathar</del><del>(</del><del>AT</del><del>)</del><del>strw</del><del>.</del><del>leidenuniv</del><del>.</del><del>nl</del><del>)</del><del>,</del><del> </del><ins>_</ins><ins>,</ins><ins> </ins>Feb 05 2009</tt></p> </div> </div> </div> </div> <div class="space4"></div> <div class=discussbar> <div class=discusshead>Discussion</div> <div class=discussnote> <div class=date>Fri Mar 30</div> <div class=time>17:39</div> <pre class=note><span class=user>OEIS Server</span>: https://oeis.org/edit/global/190</pre> </div> </div> <div class=space4></div> <div class=resultline></div> <div class="revbar"> <a href="/history/view?seq=A119812&v=4">#4</a> by <a href="/history?user=N.%20J.%20A.%20Sloane">N. J. A. Sloane</a> at Fri Feb 27 03:00:00 EST 2009 </div> <div> <div class=entry> <div class=section> <div class=sectname>DATA</div> <div class=sectbody> <p class="diffs"><tt><del>0</del><del>, </del>8, 5, 8, 2, 6, 7, 6, 5, 6, 4, 6, 1, 0, 0, 2, 0, 5, 5, 7, 9, 2, 2, 6, 0, 3, 0, 8, 4, 3, 3, 3, 7, 5, 1, 4, 8, 6, 6, 4, 9, 0, 5, 1, 9, 0, 0, 8, 3, 5, 0, 6, 7, 7, 8, 6, 6, 7, 6, 8, 4, 8, 6, 7, 8, 8, 7, 8, 4, 5, 5, 3, 7, 9, 1, 9, 1, 2, 1, 1, 1, 9, 5, 4, 8, 7, 0, 4, 9, 8, 2, 7, 6, 0, 6, 4, 3, 1, 5, 3, 1, 0, 2, 5, 2</tt></p> </div> </div> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <p class="diffs"><tt><del>1,2</del></tt></p> <p class="diffs"><tt><ins>0,1</ins></tt></p> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <p class="diffs"><tt><ins>Removed leading zero and corrected offset R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Feb 05 2009</ins></tt></p> </div> </div> </div> </div> <div class=space4></div> <div class=resultline></div> <div class="revbar"> <a href="/history/view?seq=A119812&v=3">#3</a> by <a href="/history?user=N.%20J.%20A.%20Sloane">N. J. A. Sloane</a> at Sat Nov 10 03:00:00 EST 2007 </div> <div> <div class=entry> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <p class="diffs"><tt><span title="a decimal expansion of a number">cons</span>,<span title="a sequence of nonnegative numbers">nonn</span><del>,</del><del><span title="added within the last two weeks">new</span></del></tt></p> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <p class="diffs"><tt>Paul D<del> </del><ins>.</ins><ins> </ins>Hanna (pauldhanna(AT)juno.com), May 26 2006</tt></p> </div> </div> </div> </div> <div class=space4></div> <div class=resultline></div> <div class="revbar"> <a href="/history/view?seq=A119812&v=2">#2</a> by <a href="/history?user=N.%20J.%20A.%20Sloane">N. J. A. Sloane</a> at Fri May 11 03:00:00 EDT 2007 </div> <div> <div class=entry> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <p class="diffs"><tt>Dual constant: <a href="/A119809" title="Decimal expansion of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} 1/2^A049472(n) = Sum_{n>...">A119809</a> = Sum_{n>=1} 1/2^<a href="/A049472" title="a(n) = floor(n/sqrt(2)).">A049472</a>(n) = Sum_{n>=1} <a href="/A001951" title="A Beatty sequence: a(n) = floor(n*sqrt(2)).">A001951</a>(n)/2^n. The binary expansion of this constant is given by <a href="/A080764" title="First differences of A049472, floor(n/sqrt(2)).">A080764</a> with offset n=1. Plouffe's Inverter describes approximations to this constant as "polylogarithms type of series with the floor <del>funtion</del><del> </del><ins>function</ins><ins> </ins>[ ]."</tt></p> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <p class="diffs"><tt><span title="a decimal expansion of a number">cons</span>,<span title="a sequence of nonnegative numbers">nonn</span><del>,</del><del><span title="added within the last two weeks">new</span></del></tt></p> </div> </div> </div> </div> <div class=space4></div> <div class=resultline></div> <div class="revbar"> <a href="/history/view?seq=A119812&v=1">#1</a> by <a href="/history?user=N.%20J.%20A.%20Sloane">N. J. A. Sloane</a> at Fri Sep 29 03:00:00 EDT 2006 </div> <div> <div class=entry> <div class=section> <div class=sectname>NAME</div> <div class=sectbody> <p class="diffs"><tt><ins>Decimal expansion of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} <a href="/A049472" title="a(n) = floor(n/sqrt(2)).">A049472</a>(n)/2^n = Sum_{n>=1} 1/2^<a href="/A001951" title="A Beatty sequence: a(n) = floor(n*sqrt(2)).">A001951</a>(n).</ins></tt></p> </div> </div> <div class=section> <div class=sectname>DATA</div> <div class=sectbody> <p class="diffs"><tt><ins>0, 8, 5, 8, 2, 6, 7, 6, 5, 6, 4, 6, 1, 0, 0, 2, 0, 5, 5, 7, 9, 2, 2, 6, 0, 3, 0, 8, 4, 3, 3, 3, 7, 5, 1, 4, 8, 6, 6, 4, 9, 0, 5, 1, 9, 0, 0, 8, 3, 5, 0, 6, 7, 7, 8, 6, 6, 7, 6, 8, 4, 8, 6, 7, 8, 8, 7, 8, 4, 5, 5, 3, 7, 9, 1, 9, 1, 2, 1, 1, 1, 9, 5, 4, 8, 7, 0, 4, 9, 8, 2, 7, 6, 0, 6, 4, 3, 1, 5, 3, 1, 0, 2, 5, 2</ins></tt></p> </div> </div> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <p class="diffs"><tt><ins>1,2</ins></tt></p> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <p class="diffs"><tt><ins>Dual constant: <a href="/A119809" title="Decimal expansion of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} 1/2^A049472(n) = Sum_{n>...">A119809</a> = Sum_{n>=1} 1/2^<a href="/A049472" title="a(n) = floor(n/sqrt(2)).">A049472</a>(n) = Sum_{n>=1} <a href="/A001951" title="A Beatty sequence: a(n) = floor(n*sqrt(2)).">A001951</a>(n)/2^n. The binary expansion of this constant is given by <a href="/A080764" title="First differences of A049472, floor(n/sqrt(2)).">A080764</a> with offset n=1. Plouffe's Inverter describes approximations to this constant as "polylogarithms type of series with the floor funtion [ ]."</ins></tt></p> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <p class="diffs"><tt><ins>c = 0.858267656461002055792260308433375148664905190083506778667684867..</ins></tt></p> <p class="diffs"><tt><ins>Continued fraction (<a href="/A119813" title="Partial quotients of the continued fraction of the constant A119812 defined by binary sums involving Beatty sequences: c = S...">A119813</a>):</ins></tt></p> <p class="diffs"><tt><ins>c = [0;1,6,18,1032,16777344,288230376151842816,...]</ins></tt></p> <p class="diffs"><tt><ins>where partial quotients are given by:</ins></tt></p> <p class="diffs"><tt><ins>PQ[n] = 4^<a href="/A000129" title="Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).">A000129</a>(n-2) + 2^<a href="/A001333" title="Pell-Lucas numbers: numerators of continued fraction convergents to sqrt(2).">A001333</a>(n-3) (n>2), with PQ[1]=0, PQ[2]=1.</ins></tt></p> <p class="diffs"><tt><ins>The following are equivalent expressions for the constant:</ins></tt></p> <p class="diffs"><tt><ins>(1) Sum_{n>=1} <a href="/A049472" title="a(n) = floor(n/sqrt(2)).">A049472</a>(n)/2^n; <a href="/A049472" title="a(n) = floor(n/sqrt(2)).">A049472</a>(n)=[n/sqrt(2)];</ins></tt></p> <p class="diffs"><tt><ins>(2) Sum_{n>=1} 1/2^<a href="/A001951" title="A Beatty sequence: a(n) = floor(n*sqrt(2)).">A001951</a>(n); <a href="/A001951" title="A Beatty sequence: a(n) = floor(n*sqrt(2)).">A001951</a>(n)=[n*sqrt(2)];</ins></tt></p> <p class="diffs"><tt><ins>(3) Sum_{n>=1} <a href="/A080764" title="First differences of A049472, floor(n/sqrt(2)).">A080764</a>(n)/2^n; <a href="/A080764" title="First differences of A049472, floor(n/sqrt(2)).">A080764</a>(n)=[(n+1)/sqrt(2)]-[n/sqrt(2)];</ins></tt></p> <p class="diffs"><tt><ins>where [x] = floor(x).</ins></tt></p> <p class="diffs"><tt><ins>These series illustrate the above expressions:</ins></tt></p> <p class="diffs"><tt><ins>(1) c = 0/2^1 + 1/2^2 + 2/2^3 + 2/2^4 + 3/2^5 + 4/2^6 + 4/2^7 +...</ins></tt></p> <p class="diffs"><tt><ins>(2) c = 1/2^1 + 1/2^2 + 1/2^4 + 1/2^5 + 1/2^7 + 1/2^8 + 1/2^9 +...</ins></tt></p> <p class="diffs"><tt><ins>(3) c = 1/2^1 + 1/2^2 + 0/2^3 + 1/2^4 + 1/2^5 + 0/2^6 + 1/2^7 +...</ins></tt></p> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <p class="diffs"><tt><ins>(PARI) {a(n)=local(t=sqrt(2)/2, x=sum(m=1, 10*n, floor(m*t)/2^m)); floor(10^n*x)%10}</ins></tt></p> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <p class="diffs"><tt><ins>Cf. <a href="/A119813" title="Partial quotients of the continued fraction of the constant A119812 defined by binary sums involving Beatty sequences: c = S...">A119813</a> (continued fraction), <a href="/A119814" title="Numerators of the convergents to the continued fraction for the constant A119812 defined by binary sums involving Beatty seq...">A119814</a> (convergents); <a href="/A119809" title="Decimal expansion of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} 1/2^A049472(n) = Sum_{n>...">A119809</a> (dual constant); <a href="/A000129" title="Pell numbers: a(0) = 0, a(1) = 1; for n > 1, a(n) = 2*a(n-1) + a(n-2).">A000129</a> (Pell), <a href="/A001333" title="Pell-Lucas numbers: numerators of continued fraction convergents to sqrt(2).">A001333</a>; Beatty sequences: <a href="/A049472" title="a(n) = floor(n/sqrt(2)).">A049472</a>, <a href="/A001951" title="A Beatty sequence: a(n) = floor(n*sqrt(2)).">A001951</a>, <a href="/A080764" title="First differences of A049472, floor(n/sqrt(2)).">A080764</a>; variants: <a href="/A014565" title="Decimal expansion of rabbit constant.">A014565</a> (rabbit constant), <a href="/A073115" title="Decimal expansion of sum(k>=0, 1/2^floor(k*phi) ) where phi = (1+sqrt(5))/2.">A073115</a>.</ins></tt></p> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <p class="diffs"><tt><ins><span title="a decimal expansion of a number">cons</span>,<span title="a sequence of nonnegative numbers">nonn</span></ins></tt></p> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <p class="diffs"><tt><ins>Paul D Hanna (pauldhanna(AT)juno.com), May 26 2006</ins></tt></p> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <p class="diffs"><tt><ins>approved</ins></tt></p> </div> </div> </div> </div> <div class=space4></div> <div class=space10></div> </div> </div></div> <p> <div class=footerpad></div> <div class=footer> <center> <div class=bottom> <div class=linksbar> <a href="/">Lookup</a> <a href="/wiki/Welcome"><font color="red">Welcome</font></a> <a href="/wiki/Main_Page"><font color="red">Wiki</font></a> <a href="/wiki/Special:RequestAccount">Register</a> <a href="/play.html">Music</a> <a href="/plot2.html">Plot 2</a> <a href="/demo1.html">Demos</a> <a href="/wiki/Index_to_OEIS">Index</a> <a href="/webcam">WebCam</a> <a href="/Submit.html">Contribute</a> <a href="/eishelp2.html">Format</a> <a href="/wiki/Style_Sheet">Style Sheet</a> <a href="/transforms.html">Transforms</a> <a href="/ol.html">Superseeker</a> <a href="/recent">Recents</a> </div> <div class=linksbar> <a href="/community.html">The OEIS Community</a> </div> <div class=linksbar> Maintained by <a href="http://oeisf.org">The OEIS Foundation Inc.</a> </div> <div class=dbinfo>Last modified April 7 03:26 EDT 2025. 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