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A067727 - 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>A067727 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A067727" 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=%2fA067727">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="A067727 - 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> A067727 </div> <div class=seqname> a(n) = 7*n^2 + 14*n. </div> </div> <div class=scorerefs> 9 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>21, 56, 105, 168, 245, 336, 441, 560, 693, 840, 1001, 1176, 1365, 1568, 1785, 2016, 2261, 2520, 2793, 3080, 3381, 3696, 4025, 4368, 4725, 5096, 5481, 5880, 6293, 6720, 7161, 7616, 8085, 8568, 9065, 9576, 10101, 10640, 11193, 11760, 12341, 12936</div> <div class=seqdatalinks> (<a href="/A067727/list">list</a>; <a href="/A067727/graph">graph</a>; <a href="/search?q=A067727+-id:A067727">refs</a>; <a href="/A067727/listen">listen</a>; <a href="/history?seq=A067727">history</a>; <a href="/search?q=id:A067727&fmt=text">text</a>; <a href="/A067727/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>COMMENTS</div> <div class=sectbody> <div class=sectline>Positive numbers k such that 7*(7 + k) is a perfect square.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Vincenzo Librandi, <a href="/A067727/b067727.txt">Table of n, a(n) for n = 1..1000</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(n-1) - 3*a(n-2) + a(n-3). - <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Jul 08 2012</div> <div class=sectline>G.f.: 7*x*(3-x)/(1-x)^3. - <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Jul 08 2012</div> <div class=sectline>E.g.f.: 7*x*(3 + x)*exp(x). - <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Sep 01 2019</div> <div class=sectline>From <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Feb 25 2022: (Start)</div> <div class=sectline>Sum_{n&gt;=1} 1/a(n) = 3/28.</div> <div class=sectline>Sum_{n&gt;=1} (-1)^(n+1)/a(n) = 1/28. (End)</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline>seq(7*n*(n+2), n=1..45); # <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[15000], IntegerQ[ Sqrt[ 7(7 + # )]] &amp; ]</div> <div class=sectline>CoefficientList[Series[7*(3-x)/(1-x)^3, {x, 0, 50}], x] (* <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Jul 08 2012 *)</div> <div class=sectline>7*(Range[2, 45]^2 -1) (* <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Sep 01 2019 *)</div> <div class=sectline>LinearRecurrence[{3, -3, 1}, {21, 56, 105}, 50] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Dec 07 2022 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) a(n)= 7*n*(n+2) \\ <a href="/wiki/User:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Dec 07 2011</div> <div class=sectline>(Magma) [7*n*(n+2): n in [1..50]]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Jul 08 2012</div> <div class=sectline>(Sage) [7*n*(n+2) for n in (1..45)] # <a href="/wiki/User:G._C._Greubel">G. C. Greubel</a>, Sep 01 2019</div> <div class=sectline>(GAP) List([1..45], n-&gt; 7*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="/A186029" title="a(n) = n*(7*n+3)/2.">A186029</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="/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="/A067725" title="a(n) = 3*n^2 + 6*n.">A067725</a> (k=3), <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="/A301607" title="Number of nX7 0..1 arrays with every element equal to 0, 2 or 3 horizontally or vertically adjacent elements, with upper lef...">A301607</a> <a href="/A145719" title="Erroneous version of A031963.">A145719</a> <a href="/A031963" title="a(n) = number of elements of order n in simple group L_2(7) of order 168.">A031963</a> * <a href="/A254144" title="a(n) = 1*6^n + 2*5^n + 3*4^n + 4*3^n + 5*2^n + 6*1^n.">A254144</a> <a href="/A165237" title="Long legs of primitive Pythagorean triples (a,b,c) for which 2a+1, 2b+1 and 2c+1 are primes.">A165237</a> <a href="/A271734" title="Number of set partitions of [n] with maximal block length multiplicity equal to five.">A271734</a></div> <div class=sectline>Adjacent sequences: <a href="/A067724" title="a(n) = 5*n^2 + 10*n.">A067724</a> <a href="/A067725" title="a(n) = 3*n^2 + 6*n.">A067725</a> <a href="/A067726" title="a(n) = 6*n^2 + 12*n.">A067726</a> * <a href="/A067728" title="a(n) = 2*n^2 + 8*n.">A067728</a> <a href="/A067729" title="Numbers k such that floor(log(3)^k) is prime.">A067729</a> <a href="/A067730" title="Numbers n such that sigma(n-1) + sigma(n+1) = sigma(2n).">A067730</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:Charles_R_Greathouse_IV">Charles R Greathouse IV</a>, Jul 25 2010</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 8 05:33 EDT 2025. Contains 382574 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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