CINXE.COM
A060300 - 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>A060300 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A060300" 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=%2fA060300">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="A060300 - 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> A060300 </div> <div class=seqname> a(n) = (2*n*(n+1))^2. </div> </div> <div class=scorerefs> 10 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 16, 144, 576, 1600, 3600, 7056, 12544, 20736, 32400, 48400, 69696, 97344, 132496, 176400, 230400, 295936, 374544, 467856, 577600, 705600, 853776, 1024144, 1218816, 1440000, 1690000, 1971216, 2286144, 2637376, 3027600, 3459600, 3936256, 4460544, 5035536, 5664400</div> <div class=seqdatalinks> (<a href="/A060300/list">list</a>; <a href="/A060300/graph">graph</a>; <a href="/search?q=A060300+-id:A060300">refs</a>; <a href="/A060300/listen">listen</a>; <a href="/history?seq=A060300">history</a>; <a href="/search?q=id:A060300&fmt=text">text</a>; <a href="/A060300/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,2</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Arises from middle column of following triangle: 4^2, 12^2, 24^2,...:</div> <div class=sectline>....................... 3^2 + 4^2 = 5^2</div> <div class=sectline>............... 10^2 + 11^2 + 12^2 = 13^2 + 14^2</div> <div class=sectline>........ 21^2 + 22^2 + 23^2 + 24^2 = 25^2 + 26^2 + 27^2</div> <div class=sectline>. 36^2 + 37^2 + 38^2 + 39^2 + 40^2 = 41^2 + 42^2 + 43^2 + 44^2, etc.</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>C. Stanley Ogilvy and John T. Anderson, Excursions in Number Theory, Oxford University Press, NY, 1966, pp. 90-92.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Harry J. Smith, <a href="/A060300/b060300.txt">Table of n, a(n) for n = 0..1000</a></div> <div class=sectline><a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (5, -10, 10, -5, 1).</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>G.f.: 16*x*(1+4*x+x^2)/(1-x)^5. - <a href="/wiki/User:Colin_Barker">Colin Barker</a>, Apr 22 2012</div> <div class=sectline>a(n) = 4*<a href="/A035287" title="Number of ways to place a non-attacking white and black rook on n X n chessboard.">A035287</a>(n+1) = 4*<a href="/A002378" title="Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).">A002378</a>(n)^2. - <a href="/wiki/User:Michel_Marcus">Michel Marcus</a>, May 24 2016</div> <div class=sectline>a(n) = 16 * <a href="/A000537" title="Sum of first n cubes; or n-th triangular number squared.">A000537</a>(n) = 16 * (n*(n+1)/2)^2 = 16 * <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(n)^2 = <a href="/A046092" title="4 times triangular numbers: a(n) = 2*n*(n+1).">A046092</a>(n)^2. - <a href="/wiki/User:Bruce_J._Nicholson">Bruce J. Nicholson</a>, Jun 05 2017</div> <div class=sectline>a(n) = Integral_{x=1..2*n+1} (x^3-x) dx. - <a href="/wiki/User:C茅sar_Aguilera">C茅sar Aguilera</a>, Jun 27 2020</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>CoefficientList[Series[16 x (1 + 4 x + x^2) / (1 - x)^5, {x, 0, 33}], x] (* <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Nov 18 2016 *)</div> <div class=sectline>Table[(2n(n+1))^2, {n, 0, 30}] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Jan 19 2019 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) a(n) = { (2*n*(n + 1))^2 } \\ <a href="/wiki/User:Harry_J._Smith">Harry J. Smith</a>, Jul 03 2009</div> <div class=sectline>(Magma) [(2*n*(n+1))^2: n in [0..30]]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Nov 18 2016</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>, <a href="/A000537" title="Sum of first n cubes; or n-th triangular number squared.">A000537</a>, <a href="/A002378" title="Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).">A002378</a>, <a href="/A035287" title="Number of ways to place a non-attacking white and black rook on n X n chessboard.">A035287</a>, <a href="/A046092" title="4 times triangular numbers: a(n) = 2*n*(n+1).">A046092</a>.</div> <div class=sectline>Sequence in context: <a href="/A017114" title="a(n) = (8*n + 4)^2.">A017114</a> <a href="/A331741" title="Squares s such that A331733(s) = sigma(A225546(n)) is congruent to 2 modulo 4.">A331741</a> <a href="/A092820" title="a(0) = 1; for n>0, a(n) = 16 times sum of cubes of divisors of n.">A092820</a> * <a href="/A128985" title="a(n) = (n^3 - n^2)*2^n.">A128985</a> <a href="/A341369" title="Expansion of (1 / theta_4(x) - 1)^8 / 256.">A341369</a> <a href="/A004409" title="Expansion of ( Sum_{n = -infinity..infinity} x^(n^2) )^(-8).">A004409</a></div> <div class=sectline>Adjacent sequences: <a href="/A060297" title="Periodic part of decimal expansion of n / next prime > n.">A060297</a> <a href="/A060298" title="Number of powers x^y (x,y > 1) with n digits.">A060298</a> <a href="/A060299" title="Triangle T(m,n), 1 <= n <= m, read by rows: row m lists the first m consecutive composite numbers that are not in previous rows.">A060299</a> * <a href="/A060301" title="For n>3: a(n) is a multiple of three distinct earlier terms.">A060301</a> <a href="/A060302" title="Decimal expansion of (Pi^4 + Pi^5)^(1/6).">A060302</a> <a href="/A060303" title="Number of primes below n^2 does not exceed n times the number of primes below n.">A060303</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:Jason_Earls">Jason Earls</a>, Mar 25 2001</div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Corrected the definition from 2n(n+1)^2 to (2n(n+1))^2. - <a href="/wiki/User:Harry_J._Smith">Harry J. Smith</a>, Jul 03 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 March 2 05:25 EST 2025. Contains 381316 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>