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A067725 - 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>A067725 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A067725" 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=%2fA067725">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="A067725 - 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> A067725 </div> <div class=seqname> a(n) = 3*n^2 + 6*n. </div> </div> <div class=scorerefs> 23 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 9, 24, 45, 72, 105, 144, 189, 240, 297, 360, 429, 504, 585, 672, 765, 864, 969, 1080, 1197, 1320, 1449, 1584, 1725, 1872, 2025, 2184, 2349, 2520, 2697, 2880, 3069, 3264, 3465, 3672, 3885, 4104, 4329, 4560, 4797, 5040, 5289, 5544, 5805, 6072, 6345, 6624</div> <div class=seqdatalinks> (<a href="/A067725/list">list</a>; <a href="/A067725/graph">graph</a>; <a href="/search?q=A067725+-id:A067725">refs</a>; <a href="/A067725/listen">listen</a>; <a href="/history?seq=A067725">history</a>; <a href="/search?q=id:A067725&fmt=text">text</a>; <a href="/A067725/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>Numbers h such that 3*(3 + h) is a perfect square. - <a href="/wiki/User:Alexander_D._Healy">Alexander D. Healy</a>, Tj Tullo, Avery Pickford, Sep 20 2004</div> <div class=sectline>Equivalently, numbers k such that k/3+1 is a square. - <a href="/wiki/User:Bruno_Berselli">Bruno Berselli</a>, Apr 10 2018</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>G. C. Greubel, <a href="/A067725/b067725.txt">Table of n, a(n) for n = 0..5000</a></div> <div class=sectline><a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,1).</div> </div> </div> <div class=section> <div class=sectname>FORMULA</div> <div class=sectbody> <div class=sectline>a(n) = 3*<a href="/A005563" title="a(n) = n*(n+2) = (n+1)^2 - 1.">A005563</a>(n). - <a href="/wiki/User:Zerinvary_Lajos">Zerinvary Lajos</a>, Mar 06 2007</div> <div class=sectline>a(n) = a(n-1) + 6*n + 3, with n>0, a(0)=0. - <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Aug 08 2010</div> <div class=sectline>From <a href="/wiki/User:Colin_Barker">Colin Barker</a>, Apr 11 2012: (Start)</div> <div class=sectline>a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).</div> <div class=sectline>G.f.: 3*x*(3-x)/(1-x)^3. (End)</div> <div class=sectline>E.g.f.: 3*x*(x + 3)*exp(x). - <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Jul 20 2017</div> <div class=sectline>From <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Feb 26 2022: (Start)</div> <div class=sectline>Sum_{n>=1} 1/a(n) = 1/4.</div> <div class=sectline>Sum_{n>=1} (-1)^(n+1)/a(n) = 1/12. (End)</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>seq(3*n*(n+2), n=0..50); # <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Sep 01 2019</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Select[ Range[10000], IntegerQ[ Sqrt[ 3(3 + # )]] & ]</div> <div class=sectline>3*(Range[50]^2 -1) (* <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Sep 01 2019 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) a(n)=3*n*(n+2) \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Dec 07 2011</div> <div class=sectline>(Magma) [3*n*(n+2): n in [0..50]]; // <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Sep 01 2019</div> <div class=sectline>(Sage) [3*n*(n+2) for n in (0..50)] # <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Sep 01 2019</div> <div class=sectline>(GAP) List([0..50], n-> 3*n*(n+2)); # <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Sep 01 2019</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A005563" title="a(n) = n*(n+2) = (n+1)^2 - 1.">A005563</a>.</div> <div class=sectline>Cf. numbers k such that k*(k + m) is a perfect square: <a href="/A028560" title="a(n) = n*(n + 6).">A028560</a> (k=9), <a href="/A067728" title="a(n) = 2*n^2 + 8*n.">A067728</a> (k=8), <a href="/A067727" title="a(n) = 7*n^2 + 14*n.">A067727</a> (k=7), <a href="/A067726" title="a(n) = 6*n^2 + 12*n.">A067726</a> (k=6), <a href="/A067724" title="a(n) = 5*n^2 + 10*n.">A067724</a> (k=5), <a href="/A028347" title="a(n) = n^2 - 4.">A028347</a> (k=4), <a href="/A054000" title="a(n) = 2*n^2 - 2.">A054000</a> (k=2), <a href="/A005563" title="a(n) = n*(n+2) = (n+1)^2 - 1.">A005563</a> (k=1).</div> <div class=sectline>Sequence in context: <a href="/A262044" title="Partial sum of the first n odd composite numbers.">A262044</a> <a href="/A362420" title="Partial sum of the first n odd semiprimes.">A362420</a> <a href="/A097658" title="Duplicate of A067725.">A097658</a> * <a href="/A213903" title="Least number x such that there are n numbers of the form 6k-1 or 6k+1 between prime(x) and prime(x+1).">A213903</a> <a href="/A351043" title="Lexicographically earliest non-extendable Racetrack trajectory (using von Neumann neighborhood) on spiral on infinite square...">A351043</a> <a href="/A001106" title="9-gonal (or enneagonal or nonagonal) numbers: a(n) = n*(7*n-5)/2.">A001106</a></div> <div class=sectline>Adjacent sequences: <a href="/A067722" title="Least positive integer k such that n*(n + k) is a perfect square.">A067722</a> <a href="/A067723" title="Nontrivial palindromes k such that phi(k) is also a palindrome.">A067723</a> <a href="/A067724" title="a(n) = 5*n^2 + 10*n.">A067724</a> * <a href="/A067726" title="a(n) = 6*n^2 + 12*n.">A067726</a> <a href="/A067727" title="a(n) = 7*n^2 + 14*n.">A067727</a> <a href="/A067728" title="a(n) = 2*n^2 + 8*n.">A067728</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:Robert_G._Wilson_v">Robert G. Wilson v</a>, Feb 05 2002</div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>Edited by <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Sep 14 2008 at the suggestion of <a href="/wiki/User:R._J._Mathar">R. J. Mathar</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 April 7 18:01 EDT 2025. 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