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A119811 - 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>A119811 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A119811" 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=%2fA119811">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="A119811 - 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> A119811 </div> <div class=seqname> Numerators of the convergents to the continued fraction for the 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> defined by binary sums involving Beatty sequences: c = 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. </div> </div> <div class=scorerefs> 3 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>2, 7, 72, 9511, 1246930216, 2742028548141904733479, 1737967067447512977484869808775151193351704374584616</div> <div class=seqdatalinks> (<a href="/A119811/list">list</a>; <a href="/A119811/graph">graph</a>; <a href="/search?q=A119811+-id:A119811">refs</a>; <a href="/A119811/listen">listen</a>; <a href="/history?seq=A119811">history</a>; <a href="/search?q=id:A119811&fmt=text">text</a>; <a href="/A119811/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,1</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>The number of digits in these numerators are (beginning at n=1): [1,1,2,4,10,22,52,124,297,717,1729,4173,10074,24319,58709,141735,..].</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A119811/b119811.txt">Table of n, a(n) for n=1..7.</a></div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>c = 2.32258852258806773012144068278798408011950250800432925665718...</div> <div class=sectline>Convergents begin:</div> <div class=sectline>[2/1, 7/3, 72/31, 9511/4095, 1246930216/536870911,...]</div> <div class=sectline>where the denominators of the convergents equal [2^<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-1)-1]:</div> <div class=sectline>[1,3,31,4095,536870911,1180591620717411303423,...],</div> <div class=sectline>and <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> is the Pell numbers.</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) {a(n)=local(M=contfracpnqn(vector(n, k, if(k==1, 2, 2^round(((1+sqrt(2))^(k-1)+(1-sqrt(2))^(k-1))/2) +2^round(((1+sqrt(2))^(k-2)-(1-sqrt(2))^(k-2))/(2*sqrt(2))))))); return(M[1, 1])}</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <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> (constant), <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> (continued fraction), <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>; <a href="/A119812" title="Decimal expansion of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} A049472(n)/2^n = Sum_{n>...">A119812</a> (dual constant).</div> <div class=sectline>Sequence in context: <a href="/A141315" title="INVERTi transform of A141314.">A141315</a> <a href="/A215637" title="Number of solutions of square array of integers, choosing one prime from each row and column.">A215637</a> <a href="/A381373" title="Sum over all partitions of [n] of n^j for a partition with j inversions.">A381373</a> * <a href="/A319621" title="Number of non-isomorphic connected antichain covers of n vertices by distinct sets whose dual is also an antichain of (not n...">A319621</a> <a href="/A167526" title="Prime factors of the speed of light in a vacuum, c = 299792458 (m/s).">A167526</a> <a href="/A064646" title="Numerators of partial sums of reciprocals of primorial numbers.">A064646</a></div> <div class=sectline>Adjacent sequences: <a href="/A119808" title="Triangle read by rows: T(n,k) is the number of ternary words of length n having k runs of consecutive 0's (0<=k<=ceiling(n/2)).">A119808</a> <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="/A119812" title="Decimal expansion of the constant defined by binary sums involving Beatty sequences: c = Sum_{n>=1} A049472(n)/2^n = Sum_{n>...">A119812</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></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><span title="numerators or denominators of sequence of rationals">frac</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>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 5 09:04 EDT 2025. Contains 382466 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>