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A085787 - 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>A085787 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A085787" 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=%2fA085787">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="A085787 - 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> A085787 </div> <div class=seqname> Generalized heptagonal numbers: m*(5*m - 3)/2, m = 0, +-1, +-2 +-3, ... </div> </div> <div class=scorerefs> 87 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 1, 4, 7, 13, 18, 27, 34, 46, 55, 70, 81, 99, 112, 133, 148, 172, 189, 216, 235, 265, 286, 319, 342, 378, 403, 442, 469, 511, 540, 585, 616, 664, 697, 748, 783, 837, 874, 931, 970, 1030, 1071, 1134, 1177, 1243, 1288, 1357, 1404, 1476, 1525, 1600, 1651, 1729</div> <div class=seqdatalinks> (<a href="/A085787/list">list</a>; <a href="/A085787/graph">graph</a>; <a href="/search?q=A085787+-id:A085787">refs</a>; <a href="/A085787/listen">listen</a>; <a href="/history?seq=A085787">history</a>; <a href="/search?q=id:A085787&fmt=text">text</a>; <a href="/A085787/internal">internal format</a>) </div> </div> </div> <div class=entry> <div class=section> <div class=sectname>OFFSET</div> <div class=sectbody> <div class=sectline>0,3</div> </div> </div> <div class=section> <div class=sectname>COMMENTS</div> <div class=sectbody> <div class=sectline>Zero together with the partial sums of <a href="/A080512" title="a(n) = n if n is odd, a(n) = 3*n/2 if n is even.">A080512</a>. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Sep 10 2011</div> <div class=sectline>Second heptagonal numbers (<a href="/A147875" title="Second heptagonal numbers: a(n) = n*(5*n+3)/2.">A147875</a>) and positive terms of <a href="/A000566" title="Heptagonal numbers (or 7-gonal numbers): n*(5*n-3)/2.">A000566</a> interleaved. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Aug 04 2012</div> <div class=sectline>These numbers appear in a theta function identity. See the Hardy-Wright reference, Theorem 355 on p. 284. See the g.f. of <a href="/A113429" title="Expansion of f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function.">A113429</a>. - <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Oct 28 2016</div> <div class=sectline>Characteristic function is <a href="/A133100" title="Expansion of f(x, x^4) in powers of x where f(, ) is Ramanujan's general theta function.">A133100</a>. - <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Jan 30 2017</div> <div class=sectline>40*a(n) + 9 is a square. - <a href="/wiki/User:Bruno_Berselli">Bruno Berselli</a>, Apr 18 2018</div> <div class=sectline>Numbers k such that the concatenation k225 is a square. - <a href="/wiki/User:Bruno_Berselli">Bruno Berselli</a>, Nov 07 2018</div> <div class=sectline>The sequence terms occur as exponents in the expansion of Sum_{n >= 0} q^(n*(n+1)) * Product_{k >= n+1} 1 - q^k = 1 - q - q^4 + q^7 + q^13 - q^18 - q^27 + + - - ... (see Hardy and Wright, Theorem 363, p. 290). - <a href="/wiki/User:Peter_Bala">Peter Bala</a>, Dec 15 2024</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, p. 284.</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>Vincenzo Librandi, <a href="/A085787/b085787.txt">Table of n, a(n) for n = 0..10000</a></div> <div class=sectline>Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, <a href="https://arxiv.org/abs/2312.05145">Cyclic permutations avoiding patterns in both one-line and cycle forms</a>, arXiv:2312.05145 [math.CO], 2023. See p. 2.</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>a(n) = <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(n) + <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(floor(n/2)).</div> <div class=sectline>a(2*n-1) = <a href="/A000566" title="Heptagonal numbers (or 7-gonal numbers): n*(5*n-3)/2.">A000566</a>(n).</div> <div class=sectline>a(2*n) = <a href="/A147875" title="Second heptagonal numbers: a(n) = n*(5*n+3)/2.">A147875</a>(n). - <a href="/wiki/User:Bruno_Berselli">Bruno Berselli</a>, Apr 18 2018</div> <div class=sectline>G.f.: x * (1 + 3*x + x^2) / ((1 - x) * (1 - x^2)^2). a(n) = a(-1-n) for all n in Z. - <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Oct 17 2006</div> <div class=sectline>a(n) = 5*n*(n + 1)/8 - 1/16 + (-1)^n*(2*n + 1)/16. - <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Jun 29 2009</div> <div class=sectline>a(n) = (<a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(n) + <a href="/A001082" title="Generalized octagonal numbers: k*(3*k-2), k=0, +- 1, +- 2, +-3, ...">A001082</a>(n))/2 = (<a href="/A001318" title="Generalized pentagonal numbers: m*(3*m - 1)/2, m = 0, +-1, +-2, +-3, ....">A001318</a>(n) + <a href="/A118277" title="Generalized 9-gonal (or enneagonal) numbers: m*(7*m - 5)/2 with m = 0, 1, -1, 2, -2, 3, -3, ...">A118277</a>(n))/2. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Jan 11 2013</div> <div class=sectline>a(n) = <a href="/A002378" title="Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).">A002378</a>(n) - <a href="/A001318" title="Generalized pentagonal numbers: m*(3*m - 1)/2, m = 0, +-1, +-2, +-3, ....">A001318</a>(n). - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Oct 23 2013</div> <div class=sectline>Sum_{n>=1} 1/a(n) = 10/9 + (2*sqrt(1 - 2/sqrt(5))*Pi)/3. - <a href="/wiki/User:Vaclav_Kotesovec">Vaclav Kotesovec</a>, Oct 05 2016</div> <div class=sectline>E.g.f.: (x*(9 + 5*x)*exp(x) - (1 - 2*x)*sinh(x))/8. - <a href="/wiki/User:Franck_Maminirina_Ramaharo">Franck Maminirina Ramaharo</a>, Nov 07 2018</div> <div class=sectline>Sum_{n>=1} (-1)^(n+1)/a(n) = 5*log(5)/3 - 10/9 - 2*sqrt(5)*log(phi)/3, where phi is the golden ratio (<a href="/A001622" title="Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2.">A001622</a>). - <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Feb 28 2022</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>From the first formula: a(5) = <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(5) + <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(2) = 15 + 3 = 18.</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Select[Table[(n*(n+1)/2-1)/5, {n, 500}], IntegerQ] (* <a href="/wiki/User:Vladimir_Joseph_Stephan_Orlovsky">Vladimir Joseph Stephan Orlovsky</a>, Feb 06 2012 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) t(n)=n*(n+1)/2</div> <div class=sectline>for(i=0, 40, print1(t(i)+t(floor(i/2)), ", "))</div> <div class=sectline>(PARI) {a(n) = (5*(-n\2)^2 - (-n\2)*3*(-1)^n) / 2}; /* <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Oct 17 2006 */</div> <div class=sectline>(Magma) [5*n*(n+1)/8-1/16+(-1)^n*(2*n+1)/16: n in [0..60]]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Sep 11 2011</div> <div class=sectline>(Haskell)</div> <div class=sectline>a085787 n = a085787_list !! n</div> <div class=sectline>a085787_list = scanl (+) 0 a080512_list</div> <div class=sectline>-- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Apr 06 2015</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Column 3 of <a href="/A195152" title="Square array read by antidiagonals with T(n,k) = n*((k+2)*n-k)/2, n=0, +- 1, +- 2,..., k>=0.">A195152</a>.</div> <div class=sectline>Sequences of generalized k-gonal numbers: <a href="/A001318" title="Generalized pentagonal numbers: m*(3*m - 1)/2, m = 0, +-1, +-2, +-3, ....">A001318</a> (k=5), <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a> (k=6), this sequence (k=7), <a href="/A001082" title="Generalized octagonal numbers: k*(3*k-2), k=0, +- 1, +- 2, +-3, ...">A001082</a> (k=8), <a href="/A118277" title="Generalized 9-gonal (or enneagonal) numbers: m*(7*m - 5)/2 with m = 0, 1, -1, 2, -2, 3, -3, ...">A118277</a> (k=9), <a href="/A074377" title="Generalized 10-gonal numbers: m*(4*m - 3) for m = 0, +- 1, +- 2, +- 3, ...">A074377</a> (k=10), <a href="/A195160" title="Generalized 11-gonal (or hendecagonal) numbers: m*(9*m - 7)/2 with m = 0, 1, -1, 2, -2, 3, -3, ...">A195160</a> (k=11), <a href="/A195162" title="Generalized 12-gonal numbers: k*(5*k-4) for k = 0, +-1, +-2, ...">A195162</a> (k=12), <a href="/A195313" title="Generalized 13-gonal numbers: m*(11*m-9)/2 with m = 0, 1, -1, 2, -2, 3, -3, ...">A195313</a> (k=13), <a href="/A195818" title="Generalized 14-gonal numbers: m*(6*m-5), m = 0,+1,-1,+2,-2,+3,-3,...">A195818</a> (k=14), <a href="/A277082" title="Generalized 15-gonal (or pentadecagonal) numbers: n*(13*n - 11)/2, n = 0,+1,-1,+2,-2,+3,-3, ...">A277082</a> (k=15), <a href="/A274978" title="Integers of the form m*(m + 6)/7.">A274978</a> (k=16), <a href="/A303305" title="Generalized 17-gonal (or heptadecagonal) numbers: m*(15*m - 13)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...">A303305</a> (k=17), <a href="/A274979" title="Integers of the form m*(m + 7)/8.">A274979</a> (k=18), <a href="/A303813" title="Generalized 19-gonal (or enneadecagonal) numbers: m*(17*m - 15)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...">A303813</a> (k=19), <a href="/A218864" title="Numbers of the form 9*k^2 + 8*k, k an integer.">A218864</a> (k=20), <a href="/A303298" title="Generalized 21-gonal (or icosihenagonal) numbers: m*(19*m - 17)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...">A303298</a> (k=21), <a href="/A303299" title="Generalized 22-gonal (or icosidigonal) numbers: m*(10*m - 9) with m = 0, +1, -1, +2, -2, +3, -3, ...">A303299</a> (k=22), <a href="/A303303" title="Generalized 23-gonal (or icositrigonal) numbers: m*(21*m - 19)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...">A303303</a> (k=23), <a href="/A303814" title="Generalized 24-gonal (or icositetragonal) numbers: m*(11*m - 10) with m = 0, +1, -1, +2, -2, +3, -3, ...">A303814</a> (k=24), <a href="/A303304" title="Generalized 25-gonal (or icosipentagonal) numbers: m*(23*m - 21)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...">A303304</a> (k=25), <a href="/A316724" title="Generalized 26-gonal (or icosihexagonal) numbers: m*(12*m - 11) with m = 0, +1, -1, +2, -2, +3, -3, ...">A316724</a> (k=26), <a href="/A316725" title="Generalized 27-gonal (or icosiheptagonal) numbers: m*(25*m - 23)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...">A316725</a> (k=27), <a href="/A303812" title="Generalized 28-gonal (or icosioctagonal) numbers: m*(13*m - 12) with m = 0, +1, -1, +2, -2, +3, -3, ...">A303812</a> (k=28), <a href="/A303815" title="Generalized 29-gonal (or icosienneagonal) numbers: m*(27*m - 25)/2 with m = 0, +1, -1, +2, -2, +3, -3, ...">A303815</a> (k=29), <a href="/A316729" title="Generalized 30-gonal (or triacontagonal) numbers: m*(14*m - 13) with m = 0, +1, -1, +2, -2, +3, -3, ...">A316729</a> (k=30).</div> <div class=sectline>Cf. <a href="/A001622" title="Decimal expansion of golden ratio phi (or tau) = (1 + sqrt(5))/2.">A001622</a>, <a href="/A080512" title="a(n) = n if n is odd, a(n) = 3*n/2 if n is even.">A080512</a>, <a href="/A113429" title="Expansion of f(-x, -x^4) in powers of x where f(, ) is Ramanujan's general theta function.">A113429</a>, <a href="/A133100" title="Expansion of f(x, x^4) in powers of x where f(, ) is Ramanujan's general theta function.">A133100</a>.</div> <div class=sectline>Sequence in context: <a href="/A074136" title="Main diagonal of triangle A074135.">A074136</a> <a href="/A310824" title="Coordination sequence Gal.6.54.2 where Gal.u.t.v denotes the coordination sequence for a vertex of type v in tiling number t...">A310824</a> <a href="/A266811" title="Total number of ON (black) cells after n iterations of the "Rule 62" elementary cellular automaton starting with a single ON...">A266811</a> * <a href="/A111710" title="Consider the triangle shown below in which the n-th row contains the n smallest numbers greater than those in the previous r...">A111710</a> <a href="/A191138" title="Increasing sequence generated by these rules: a(1)=1, and if x is in a then 3x+1 and 4x+3 are in a.">A191138</a> <a href="/A075315" title="Pair the natural numbers such that the n-th pair is (k, k+p(n)) where k is the smallest number not occurring earlier and p(n...">A075315</a></div> <div class=sectline>Adjacent sequences: <a href="/A085784" title="Product of a prime and any number of triangular numbers.">A085784</a> <a href="/A085785" title="Start at (2n+1)/4 and iterate the map x -> x*ceiling(x); sequence gives number of steps for denominator to drop to 1 or 2; o...">A085785</a> <a href="/A085786" title="a(n) = n*(2*n^2 + n + 1)/2.">A085786</a> * <a href="/A085788" title="Partial sums of n 3-spaced triangular numbers beginning with t(3), e.g., a(2)=t(3)+t(6)=6+21=27.">A085788</a> <a href="/A085789" title="Partial sums of n 3-spaced triangular numbers beginning with t(2), e.g., a(2) = t(2) + t(5) = 3 + 15 = 18.">A085789</a> <a href="/A085790" title="Integers sorted by the sum of their divisors.">A085790</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:Jon_Perry">Jon Perry</a>, Jul 23 2003</div> </div> </div> <div class=section> <div class=sectname>EXTENSIONS</div> <div class=sectbody> <div class=sectline>New name from <a href="/wiki/User:T._D._Noe">T. D. Noe</a>, Apr 21 2006</div> <div class=sectline>Formula in sequence name added by <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, May 28 2012</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 February 26 20:10 EST 2025. 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