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A119812 - OEIS
<!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>A119812 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/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=%2fA119812">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="A119812 - 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> <div class=sequence> <div class=space1></div> <div class=line></div> <div class=seqhead> <div class=seqnumname> <div class=seqnum> A119812 </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). </div> </div> <div class=scorerefs> 8 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>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</div> <div class=seqdatalinks> (<a href="/A119812/list">list</a>; <a href="/A119812/constant">constant</a>; <a href="/A119812/graph">graph</a>; <a href="/search?q=A119812+-id:A119812">refs</a>; <a href="/A119812/listen">listen</a>; <a href="/history?seq=A119812">history</a>; <a href="/search?q=id:A119812&fmt=text">text</a>; <a href="/A119812/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,1</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>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 function [ ]."</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A119812/b119812.txt">Table of n, a(n) for n=0..103.</a></div> <div class=sectline>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.</div> <div class=sectline>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.</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>c = 0.858267656461002055792260308433375148664905190083506778667684867..</div> <div class=sectline>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>):</div> <div class=sectline>c = [0;1,6,18,1032,16777344,288230376151842816,...]</div> <div class=sectline>where partial quotients are given by:</div> <div class=sectline>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.</div> <div class=sectline>The following are equivalent expressions for the constant:</div> <div class=sectline>(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)];</div> <div class=sectline>(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)];</div> <div class=sectline>(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)];</div> <div class=sectline>where [x] = floor(x).</div> <div class=sectline>These series illustrate the above expressions:</div> <div class=sectline>(1) c = 0/2^1 + 1/2^2 + 2/2^3 + 2/2^4 + 3/2^5 + 4/2^6 + 4/2^7 +...</div> <div class=sectline>(2) c = 1/2^1 + 1/2^2 + 1/2^4 + 1/2^5 + 1/2^7 + 1/2^8 + 1/2^9 +...</div> <div class=sectline>(3) c = 1/2^1 + 1/2^2 + 0/2^3 + 1/2^4 + 1/2^5 + 0/2^6 + 1/2^7 +...</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) {a(n)=local(t=sqrt(2)/2, x=sum(m=1, 10*n, floor(m*t)/2^m)); floor(10^n*x)%10}</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>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>.</div> <div class=sectline>Sequence in context: <a href="/A377981" title="Decimal expansion of G/2 - Pi*log(2)/8, where G = A006752.">A377981</a> <a href="/A305036" title="Number of nX4 0..1 arrays with every element unequal to 0, 1, 2, 6 or 7 king-move adjacent elements, with upper left element...">A305036</a> <a href="/A357106" title="Decimal expansion of the real root of 2*x^3 + x^2 - 2.">A357106</a> * <a href="/A153799" title="Decimal expansion of 4 - Pi.">A153799</a> <a href="/A086235" title="Decimal expansion of probability that a random walk on a 7-d lattice returns to the origin.">A086235</a> <a href="/A157742" title="A006094(n+3) mod 9.">A157742</a></div> <div class=sectline>Adjacent sequences: <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> <a href="/A119810" title="Partial quotients of the continued fraction of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1...">A119810</a> <a href="/A119811" title="Numerators of the convergents to the continued fraction for the constant A119809 defined by binary sums involving Beatty seq...">A119811</a> * <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> <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> <a href="/A119815" title="Integer a(n) produces the least positive integer coefficient of x^n in the n-th iteration of g.f. A(x) where A(0)=0.">A119815</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="a decimal expansion of a number"><a href="/A119812/constant">cons</a></span>,<span title="a sequence of nonnegative numbers">nonn</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Paul_D._Hanna">Paul D. Hanna</a>, May 26 2006</div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Removed leading zero and corrected offset <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Feb 05 2009</div> </div> </div> <div class=section> <div class=sectname>STATUS</div> <div class=sectbody> <div class=sectline>approved</div> </div> </div> </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 6 00:19 EDT 2025. Contains 382505 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>