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A051047 - 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>A051047 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A051047" 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=%2fA051047">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="A051047 - OEIS"></a> </div> <div class="motdbox"> <div class="motd"> <p>Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).</p> </div> <div class="donate"> <div id="donate-button-container"> <div id="donate-button"></div> <script src="https://www.paypalobjects.com/donate/sdk/donate-sdk.js" charset="UTF-8"></script> <script> PayPal.Donation.Button({ env:'production', hosted_button_id:'SVPGSDDCJ734A', image: { src:'https://www.paypalobjects.com/en_US/i/btn/btn_donateCC_LG.gif', alt:'Donate with PayPal button', title:'PayPal - The safer, easier way to pay online!', } }).render('#donate-button'); </script> </div> <a href="https://oeisf.org/donate/"> <strong>Other ways to Give</strong> </a> </div> </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> A051047 </div> <div class=seqname> For n > 5, a(n) = 15*a(n-1) - 15*a(n-2) + a(n-3); initial terms are 1, 3, 8, 120, 1680. </div> </div> <div class=scorerefs> 4 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 3, 8, 120, 1680, 23408, 326040, 4541160, 63250208, 880961760, 12270214440, 170902040408, 2380358351280, 33154114877520, 461777249934008, 6431727384198600, 89582406128846400, 1247721958419651008, 17378525011746267720, 242051628206028097080</div> <div class=seqdatalinks> (<a href="/A051047/list">list</a>; <a href="/A051047/graph">graph</a>; <a href="/search?q=A051047+-id:A051047">refs</a>; <a href="/A051047/listen">listen</a>; <a href="/history?seq=A051047">history</a>; <a href="/search?q=id:A051047&fmt=text">text</a>; <a href="/A051047/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>1,2</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>The recurrence gives an infinite sequence of polynomials S={x,x+2,c_1(x),c_2(x),...} such that the product of any two consecutive polynomials, increased by 1, is the square of a polynomial - see the Jones reference.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Colin Barker, <a href="/A051047/b051047.txt">Table of n, a(n) for n = 1..850</a></div> <div class=sectline>Andrej Dujella and Attila Petho, <a href="https://citeseerx.ist.psu.edu/pdf/9e894b39c9b7662910b3fa48407ec6d7ecce921e">Generalization of a theorem of Baker and Davenport</a></div> <div class=sectline>B. W. Jones, <a href="http://qjmath.oxfordjournals.org/content/27/3/349.extract">A Variation of a Problem of Davenport and Diophantus</a>, Quart. J. Math. (Oxford) Ser. (2) 27, 349-353, 1976.</div> <div class=sectline><a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (15,-15,1).</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>G.f.: x*(3*x^4-44*x^3+22*x^2+12*x-1) / (x^3-15*x^2+15*x-1).</div> <div class=sectline>For n>4, a(n) = 14*a(n-1)-a(n-2)+8. - <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Mar 05 2016</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>With[{x = 1},</div> <div class=sectline>Join[{x, x + 2},</div> <div class=sectline>RecurrenceTable[{c[-1] == c[0] == 0,</div> <div class=sectline>c[k] == (4 x^2 + 8 x + 2) c[k - 1] - c[k - 2] + 4 (x + 1)}, c, {k, 1, 12}]]]</div> <div class=sectline>LinearRecurrence[{15, -15, 1}, {1, 3, 8, 120, 1680}, 22] (* <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Oct 31 2011 *)</div> <div class=sectline>Join[{1, 3}, RecurrenceTable[{a[1] == 8, a[2] == 120, a[n] == 14 a[n-1] - a[n-2] + 8}, a, {n, 20}]] (* <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Mar 05 2016 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) Vec((3*x^4-44*x^3+22*x^2+12*x-1)/(x^3-15*x^2+15*x-1)+O(x^99)) \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Oct 31 2011</div> <div class=sectline>(Magma) I:=[1, 3, 8, 120, 1680]; [n le 5 select I[n] else 14*Self(n-1)-Self(n-2)+8: n in [1..20]]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Mar 05 2016</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A051048" title="Sqrt[a(n)a(n+1)+1] of A051047.">A051048</a>. Essentially the same as <a href="/A045899" title="Numbers k such that k+1 and 3*k+1 are perfect squares.">A045899</a>.</div> <div class=sectline>Sequence in context: <a href="/A134803" title="Numbers n such that the sum of all numbers of the same parity <= n is equal to the sum of numbers of the opposite parity fro...">A134803</a> <a href="/A030063" title="Fermat's Diophantine m-tuple: 1 + the product of any two distinct terms is a square.">A030063</a> <a href="/A195568" title="Denominators a(n) of Pythagorean approximations b(n)/a(n) to 7/4.">A195568</a> * <a href="/A192629" title="Numerators of the Fermat-Euler rational Diophantine m-tuple.">A192629</a> <a href="/A245458" title="a(n) = ((prime(n)-2)!+2) mod prime(n)# (cf. A002110).">A245458</a> <a href="/A036504" title="Numerator of n^(n-1)/n!.">A036504</a></div> <div class=sectline>Adjacent sequences: <a href="/A051044" title="Odd values of the PartitionsQ function A000009.">A051044</a> <a href="/A051045" title="Number of distinct resistances possible for n resistors with resistances 1, 2, ..., n, each connected in series or parallel ...">A051045</a> <a href="/A051046" title="Numbers k for which pi(k) > k/(H_k - 3/2), where pi is the prime-counting function and H_k is the k-th harmonic number.">A051046</a> * <a href="/A051048" title="Sqrt[a(n)a(n+1)+1] of A051047.">A051048</a> <a href="/A051049" title="Number of moves needed to solve an (n+1)-ring baguenaudier if two simultaneous moves of the two end rings are counted as one.">A051049</a> <a href="/A051050" title="Numerator of the probability that the convex hull of n+2 randomly chosen points in the unit ball B^n has n+1 vertices.">A051050</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>,<span title="it is very easy to produce terms of sequence">easy</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:Eric_W._Weisstein">Eric W. Weisstein</a></div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Entry revised by <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Oct 25 2009, following correspondence with Eric Weisstein</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 December 3 16:59 EST 2024. Contains 378391 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>