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A032296 - 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>A032296 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A032296" 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=%2fA032296">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="A032296 - 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> A032296 </div> <div class=seqname> Number of aperiodic bracelets (turnover necklaces) with n beads of 5 colors. </div> </div> <div class=scorerefs> 1 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>5, 10, 30, 105, 372, 1460, 5890, 25275, 110050, 492744, 2227270, 10195070, 46989180, 218096780, 1017447736, 4768944375, 22440372240, 105966686200, 501938733550, 2384200190580, 11353290083380</div> <div class=seqdatalinks> (<a href="/A032296/list">list</a>; <a href="/A032296/graph">graph</a>; <a href="/search?q=A032296+-id:A032296">refs</a>; <a href="/A032296/listen">listen</a>; <a href="/history?seq=A032296">history</a>; <a href="/search?q=id:A032296&fmt=text">text</a>; <a href="/A032296/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>LINKS</div> <div class=sectbody> <div class=sectline><a href="/A032296/b032296.txt">Table of n, a(n) for n=1..21.</a></div> <div class=sectline>C. G. Bower, <a href="/transforms2.html">Transforms (2)</a></div> <div class=sectline>F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a></div> <div class=sectline>F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]</div> <div class=sectline>N. J. A. Sloane, <a href="/transforms.txt">Transforms</a></div> <div class=sectline><a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a></div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>MOEBIUS transform of <a href="/A032276" title="Number of bracelets (turnover necklaces) with n beads of 5 colors.">A032276</a>.</div> <div class=sectline>From <a href="/wiki/User:Herbert_Kociemba">Herbert Kociemba</a>, Nov 28 2016: (Start)</div> <div class=sectline>More generally, gf(k) is the g.f. for the number of bracelets with primitive period n and beads of k colors.</div> <div class=sectline>gf(k): Sum_{n>=1} mu(n)*( -log(1-k*x^n)/n + Sum_{i=0..2} binomial(k,i)x^(n*i)/(1-k*x^(2*n)) )/2. (End)</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>mx=40; gf[x_, k_]:=Sum[ MoebiusMu[n]*(-Log[1-k*x^n]/n+Sum[Binomial[k, i]x^(n i), {i, 0, 2}]/( 1-k x^(2n)))/2, {n, mx}]; CoefficientList[Series[gf[x, 5], {x, 0, mx}], x] (* <a href="/wiki/User:Herbert_Kociemba">Herbert Kociemba</a>, Nov 28 2016 *)</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Column 5 of <a href="/A276550" title="Array read by antidiagonals: T(n,k) = number of primitive (period n) bracelets using a maximum of k different colored beads.">A276550</a>.</div> <div class=sectline>Sequence in context: <a href="/A373568" title="Expansion of x - 1/(x - 1/(x - 1/(x - 1/(x + 1)))).">A373568</a> <a href="/A189315" title="Expansion of g.f. 5*(1-3*x+x^2)/(1-5*x+5*x^2).">A189315</a> <a href="/A056422" title="Number of primitive (period n) step cyclic shifted sequences using a maximum of five different symbols.">A056422</a> * <a href="/A052648" title="Expansion of e.g.f. 5*x/(1-x).">A052648</a> <a href="/A020995" title="Numbers k such that the sum of the digits of Fibonacci(k) is k.">A020995</a> <a href="/A174467" title="G.f.: exp( Sum_{n>=1} A174468(n)*x^n/n ) where A174468(n) = Sum_{d|n} d*sigma(n/d)*sigma(d).">A174467</a></div> <div class=sectline>Adjacent sequences: <a href="/A032293" title="Number of series-reduced dyslexic planted compound windmills with n leaves of 2 colors.">A032293</a> <a href="/A032294" title="Number of aperiodic bracelets (turnover necklaces) with n beads of 3 colors.">A032294</a> <a href="/A032295" title="Number of aperiodic bracelets (turn over necklaces) with n beads of 4 colors.">A032295</a> * <a href="/A032297" title=""EFJ" (unordered, size, labeled) transform of 2,2,2,2,...">A032297</a> <a href="/A032298" title=""EFJ" (unordered, size, labeled) transform of 2,1,1,1,...">A032298</a> <a href="/A032299" title=""EFJ" (unordered, size, labeled) transform of 1,2,3,4,...">A032299</a></div> </div> </div> <div class=section> <div class=sectname>KEYWORD</div> <div class=sectbody> <div class=sectline><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:Christian_G._Bower">Christian G. Bower</a></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 November 28 14:08 EST 2024. Contains 378204 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>