A316671 - 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>A316671 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A316671" 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=%2fA316671">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="A316671 - 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> A316671 </div> <div class=seqname> Squares visited by moving diagonally one square on a diagonally numbered board and moving to the lowest available unvisited square at each step. </div> </div> <div class=scorerefs> 6 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>1, 5, 4, 12, 11, 23, 22, 38, 37, 57, 56, 80, 79, 107, 106, 138, 137, 173, 172, 212, 211, 255, 254, 302, 301, 353, 352, 408, 407, 467, 466, 530, 529, 597, 596, 668, 667, 743, 742, 822, 821, 905, 904, 992, 991, 1083, 1082, 1178, 1177, 1277, 1276, 1380, 1379</div> <div class=seqdatalinks> (<a href="/A316671/list">list</a>; <a href="/A316671/graph">graph</a>; <a href="/search?q=A316671+-id:A316671">refs</a>; <a href="/A316671/listen">listen</a>; <a href="/history?seq=A316671">history</a>; <a href="/search?q=id:A316671&fmt=text">text</a>; <a href="/A316671/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>Board is numbered as follows:</div> <div class=sectline> 1 2 4 7 11 16 .</div> <div class=sectline> 3 5 8 12 17 .</div> <div class=sectline> 6 9 13 18 .</div> <div class=sectline> 10 14 19 .</div> <div class=sectline> 15 20 .</div> <div class=sectline> 21 .</div> <div class=sectline> .</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Dani毛l Karssen, <a href="/A316671/b316671.txt">Table of n, a(n) for n = 1..10000</a></div> <div class=sectline>Dani毛l Karssen, <a href="/A316671/a316671.svg">Figure showing the first 6 steps of the sequence</a></div> <div class=sectline><a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>From <a href="/wiki/User:Colin_Barker">Colin Barker</a>, Jul 18 2018: (Start)</div> <div class=sectline>G.f.: x*(1 + 4*x - 3*x^2 + 2*x^4) / ((1 - x)^3*(1 + x)^2).</div> <div class=sectline>a(n) = a(n-1) + 2*a(n-2) - 2*a(n-3) - a(n-4) + a(n-5) for n>5.</div> <div class=sectline>a(n) = (n^2 + n + 4)/2 for n even.</div> <div class=sectline>a(n) = (n^2 - n + 2)/2 for n odd.</div> <div class=sectline>(End)</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>CoefficientList[ Series[-(2x^4 - 3x^2 + 4x + 1)/((x - 1)^3 (x + 1)^2), {x, 0, 52}], x] (* or *)</div> <div class=sectline>LinearRecurrence[{1, 2, -2, -1, 1}, {1, 5, 4, 12, 11}, 53] (* <a href="/wiki/User:Robert_G._Wilson_v">Robert G. Wilson v</a>, Jul 18 2018 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) Vec(x*(1 + 4*x - 3*x^2 + 2*x^4) / ((1 - x)^3*(1 + x)^2) + O(x^40)) \\ <a href="/wiki/User:Colin_Barker">Colin Barker</a>, Jul 18 2018</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A316588" title="Squares visited by knight moves on a diagonally numbered board and moving to the lowest available unvisited square at each step.">A316588</a>, <a href="/A316668" title="Squares visited by king moves on a diagonally numbered board and moving to the lowest available unvisited square at each step.">A316668</a>, <a href="/A316669" title="Squares visited by queen moves on a diagonally numbered board and moving to the lowest available unvisited square at each step.">A316669</a>, <a href="/A316670" title="Squares visited by bishop moves on a diagonally numbered board and moving to the lowest available unvisited square at each step.">A316670</a>.</div> <div class=sectline>Sequence in context: <a href="/A131875" title="Triangle, A000012 * A131844 as infinite lower triangular matrices.">A131875</a> <a href="/A095871" title="Triangle read by rows: T(n,k)=(n+1)*(3*(n+1)-1)/2-k*(3*k-1)/2">A095871</a> <a href="/A284551" title="Triangular array read by rows, demonstrating that the difference between a pentagonal number (left edge of triangle) and a s...">A284551</a> * <a href="/A338157" title="Numbers that follow from the alternating series a(n) = d(1) - d(2) + d(3) -d(4) + ... + (-1)^(n+1) d(n), where d(k) denotes ...">A338157</a> <a href="/A189235" title="Expansion of (5-16*x+6*x^2+10*x^3-2*x^4)/(1-4*x+2*x^2+5*x^3-2*x^4-x^5)">A189235</a> <a href="/A019068" title="Cycle class sequence c(n) (the number of true cycles of length n in which a certain node is included) for zeolite BEA = Beta...">A019068</a></div> <div class=sectline>Adjacent sequences: <a href="/A316668" title="Squares visited by king moves on a diagonally numbered board and moving to the lowest available unvisited square at each step.">A316668</a> <a href="/A316669" title="Squares visited by queen moves on a diagonally numbered board and moving to the lowest available unvisited square at each step.">A316669</a> <a href="/A316670" title="Squares visited by bishop moves on a diagonally numbered board and moving to the lowest available unvisited square at each step.">A316670</a> * <a href="/A316672" title="Numbers k for which 120*k + 169 is a square.">A316672</a> <a href="/A316673" title="Number of paths from (0,0,0) to (n,n,n) that always move closer to (n,n,n).">A316673</a> <a href="/A316674" title="Number A(n,k) of paths from {0}^k to {n}^k that always move closer to {n}^k; square array A(n,k), n>=0, k>=0, read by antidi...">A316674</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:Dani毛l_Karssen">Dani毛l Karssen</a>, Jul 15 2018</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 10 05:25 EDT 2025. Contains 382654 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>

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A316671 - OEIS