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A001318 - 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>A001318 - OEIS</title> <link rel="search" type="application/opensearchdescription+xml" title="OEIS" href="/oeis.xml"> <script> var myURL = "\/A001318" 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=%2fA001318">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="A001318 - 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> A001318 </div> <div class=seqname> Generalized pentagonal numbers: m*(3*m - 1)/2, m = 0, +-1, +-2, +-3, .... <br><font size=-1>(Formerly M1336 N0511)</font> </div> </div> <div class=scorerefs> 273 </div> </div> <div> <div class=seqdatabox> <div class=seqdata>0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335</div> <div class=seqdatalinks> (<a href="/A001318/list">list</a>; <a href="/A001318/graph">graph</a>; <a href="/search?q=A001318+-id:A001318">refs</a>; <a href="/A001318/listen">listen</a>; <a href="/history?seq=A001318">history</a>; <a href="/search?q=id:A001318&fmt=text">text</a>; <a href="/A001318/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>Partial sums of <a href="/A026741" title="a(n) = n if n odd, n/2 if n even.">A026741</a>. - <a href="/wiki/User:Jud_McCranie">Jud McCranie</a>; corrected by <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Jul 05 2012</div> <div class=sectline>From <a href="/wiki/User:R._K._Guy">R. K. Guy</a>, Dec 28 2005: (Start)</div> <div class=sectline>"Conway's relation twixt the triangular and pentagonal numbers: Divide the triangular numbers by 3 (when you can exactly):</div> <div class=sectline>0 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 ...</div> <div class=sectline>0 - 1 2 .- .5 .7 .- 12 15 .- 22 26 .- .35 .40 .- ..51 ...</div> <div class=sectline>.....-.-.....+..+.....-..-.....+..+......-...-.......+....</div> <div class=sectline>"and you get the pentagonal numbers in pairs, one of positive rank and the other negative.</div> <div class=sectline>"Append signs according as the pair have the same (+) or opposite (-) parity.</div> <div class=sectline>"Then Euler's pentagonal number theorem is easy to remember:</div> <div class=sectline>"p(n-0) - p(n-1) - p(n-2) + p(n-5) + p(n-7) - p(n-12) - p(n-15) ++-- = 0^n</div> <div class=sectline>where p(n) is the partition function, the left side terminates before the argument becomes negative and 0^n = 1 if n = 0 and = 0 if n > 0.</div> <div class=sectline>"E.g. p(0) = 1, p(7) = p(7-1) + p(7-2) - p(7-5) - p(7-7) + 0^7 = 11 + 7 - 2 - 1 + 0 = 15."</div> <div class=sectline>(End)</div> <div class=sectline>The sequence may be used in order to compute sigma(n), as described in Euler's article. - <a href="/wiki/User:Thomas_Baruchel">Thomas Baruchel</a>, Nov 19 2003</div> <div class=sectline>Number of levels in the partitions of n + 1 with parts in {1,2}.</div> <div class=sectline>a(n) is the number of 3 X 3 matrices (symmetrical about each diagonal) M = {{a, b, c}, {b, d, b}, {c, b, a}} such that a + b + c = b + d + b = n + 2, a,b,c,d natural numbers; example: a(3) = 5 because (a,b,c,d) = (2,2,1,1), (1,2,2,1), (1,1,3,3), (3,1,1,3), (2,1,2,3). - <a href="/wiki/User:Philippe_Del茅ham">Philippe Del茅ham</a>, Apr 11 2007</div> <div class=sectline>Also numbers a(n) such that 24*a(n) + 1 = (6*m - 1)^2 are odd squares: 1, 25, 49, 121, 169, 289, 361, ..., m = 0, +-1, +-2, ... . - <a href="/wiki/User:Zak_Seidov">Zak Seidov</a>, Mar 08 2008</div> <div class=sectline>From <a href="/wiki/User:Matthew_Vandermast">Matthew Vandermast</a>, Oct 28 2008: (Start)</div> <div class=sectline>Numbers n for which <a href="/A000326" title="Pentagonal numbers: a(n) = n*(3*n-1)/2.">A000326</a>(n) is a member of <a href="/A000332" title="Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24.">A000332</a>. Cf. <a href="/A145920" title="List of numbers that are both pentagonal (A000326) and binomial coefficients C(n,4) (A000332).">A145920</a>.</div> <div class=sectline>This sequence contains all members of <a href="/A000332" title="Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24.">A000332</a> and all nonnegative members of <a href="/A145919" title="A000332(n) = a(n)*(3*a(n) - 1)/2.">A145919</a>. For values of n such that n*(3*n - 1)/2 belongs to <a href="/A000332" title="Binomial coefficient binomial(n,4) = n*(n-1)*(n-2)*(n-3)/24.">A000332</a>, see <a href="/A145919" title="A000332(n) = a(n)*(3*a(n) - 1)/2.">A145919</a>. (End)</div> <div class=sectline>Starting with offset 1 = row sums of triangle <a href="/A168258" title="Triangle read by rows, A101688 * A000012 as infinite lower triangular matrices.">A168258</a>. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Nov 21 2009</div> <div class=sectline>Starting with offset 1 = Triangle <a href="/A101688" title="Once 1, once 0, repeat, twice 1, twice 0, repeat, thrice 1, thrice 0, ... and so on.">A101688</a> * [1, 2, 3, ...]. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Nov 27 2009</div> <div class=sectline>Starting with offset 1 can be considered the first in an infinite set generated from <a href="/A026741" title="a(n) = n if n odd, n/2 if n even.">A026741</a>. Refer to the array in <a href="/A175005" title="Expansion of x/(1 - 4*x + 3*x^2 - 2*x^3).">A175005</a>. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Apr 03 2010</div> <div class=sectline>Vertex number of a square spiral whose edges have length <a href="/A026741" title="a(n) = n if n odd, n/2 if n even.">A026741</a>. The two axes of the spiral forming an "X" are <a href="/A000326" title="Pentagonal numbers: a(n) = n*(3*n-1)/2.">A000326</a> and <a href="/A005449" title="Second pentagonal numbers: a(n) = n*(3*n + 1)/2.">A005449</a>. The four semi-axes forming an "X" are <a href="/A049452" title="Pentagonal numbers with even index.">A049452</a>, <a href="/A049453" title="Second pentagonal numbers with even index: a(n) = n*(6*n+1).">A049453</a>, <a href="/A033570" title="Pentagonal numbers with odd index: a(n) = (2*n+1)*(3*n+1).">A033570</a> and the numbers >= 2 of <a href="/A033568" title="Second pentagonal numbers with odd index: a(n) = (2*n-1)*(3*n-1).">A033568</a>. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Sep 08 2011</div> <div class=sectline>A general formula for the generalized k-gonal numbers is given by n*((k - 2)*n - k + 4)/2, n=0, +-1, +-2, ..., k >= 5. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Sep 15 2011</div> <div class=sectline>a(n) is the number of 3-tuples (w,x,y) having all terms in {0,...,n} and 2*w = 2*x + y. - <a href="/wiki/User:Clark_Kimberling">Clark Kimberling</a>, Jun 04 2012</div> <div class=sectline>Generalized k-gonal numbers are second k-gonal numbers and positive terms of k-gonal numbers interleaved, k >= 5. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Aug 04 2012</div> <div class=sectline>a(n) is the sum of the largest parts of the partitions of n+1 into exactly 2 parts. - <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Jan 26 2013</div> <div class=sectline>Conway's relation mentioned by <a href="/wiki/User:R._K._Guy">R. K. Guy</a> is a relation between triangular numbers and generalized pentagonal numbers, two sequences from different families, but as triangular numbers are also generalized hexagonal numbers in this case we have a relation between two sequences from the same family. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Feb 01 2013</div> <div class=sectline>Start with the sequence of all 0's. Add n to each value of a(n) and the next n - 1 terms. The result is the generalized pentagonal numbers. - <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Nov 03 2014</div> <div class=sectline>(6k + 1) | a(4k). (3k + 1) | a(4k+1). (3k + 2) | a(4k+2). (6k + 5) | a(4k+3). - <a href="/wiki/User:Jon_Perry">Jon Perry</a>, Nov 04 2014</div> <div class=sectline>Enge, Hart and Johansson proved: "Every generalised pentagonal number c >= 5 is the sum of a smaller one and twice a smaller one, that is, there are generalised pentagonal numbers a, b < c such that c = 2a + b." (see link theorem 5). - <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, Aug 26 2016</div> <div class=sectline>The Enge, et al. result for c >= 5 also holds for c >= 2 if 0 is included as a generalized pentagonal number. That is, 2 = 2*1 + 0. - <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Jun 02 2018</div> <div class=sectline>Suggestion for title, where n actually matches the list and b-file: "Generalized pentagonal numbers: k(n)*(3*k(n) - 1)/2, where k(n) = <a href="/A001057" title="Canonical enumeration of integers: interleaved positive and negative integers with zero prepended.">A001057</a>(n) = [0, 1, -1, 2, -2, 3, -3, ...], n >= 0" - <a href="/wiki/User:Daniel_Forgues">Daniel Forgues</a>, Jun 09 2018 & Jun 12 2018</div> <div class=sectline>Generalized k-gonal numbers are the partial sums of the sequence formed by the multiples of (k - 4) and the odd numbers (<a href="/A005408" title="The odd numbers: a(n) = 2*n + 1.">A005408</a>) interleaved, with k >= 5. - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Jul 25 2018</div> <div class=sectline>The last digits form a symmetric cycle of length 40 [0, 1, 2, 5, ..., 5, 2, 1, 0], i.e., a(n) == a(n + 40) (mod 10) and a(n) == a(40*k - n - 1) (mod 10), 40*k > n. - <a href="/wiki/User:Alejandro_J._Becerra_Jr.">Alejandro J. Becerra Jr.</a>, Aug 14 2018</div> <div class=sectline>Only 2, 5, and 7 are prime. All terms are of the form k*(k+1)/6, where 3 | k or 3 | k+1. For k > 6, the value divisible by 3 must have another factor d > 2, which will remain after the division by 6. - <a href="/wiki/User:Eric_Snyder">Eric Snyder</a>, Jun 03 2022</div> <div class=sectline>8*a(n) is the product of two even numbers one of which is n + n mod 2. - <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, Jul 15 2022</div> <div class=sectline>a(n) is the dot product of [1, 2, 3, ..., n] and repeat[1, 1/2]. a(5) = 12 = [1, 2, 3, 4, 5] dot [1, 1/2, 1, 1/2, 1] = [1 + 1 + 3 + 2 + 5]. - <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Dec 10 2022</div> <div class=sectline>Every nonnegative number is the sum of four terms of this sequence [S. Realis]. - <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, May 07 2023</div> </div> </div> <div class=section> <div class=sectname>REFERENCES</div> <div class=sectbody> <div class=sectline>Enoch Haga, A strange sequence and a brilliant discovery, chapter 5 of Exploring prime numbers on your PC and the Internet, first revised ed., 2007 (and earlier ed.), pp. 53-70.</div> <div class=sectline>Ross Honsberger, Ingenuity in Mathematics, Random House, 1970, p. 117.</div> <div class=sectline>Donald E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, (to appear), section 7.2.1.4, equation (18).</div> <div class=sectline>Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, 2nd ed., Wiley, NY, 1966, p. 231.</div> <div class=sectline>N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).</div> <div class=sectline>N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).</div> </div> </div> <div class=section> <div class=sectname>LINKS</div> <div class=sectbody> <div class=sectline>T. D. Noe, <a href="/A001318/b001318.txt">Table of n, a(n) for n = 0..1000</a></div> <div class=sectline>G. E. Andrews and J. A. Sellers, <a href="http://arxiv.org/abs/1401.5345">Congruences for the Fishburn Numbers</a>, arXiv preprint arXiv:1401.5345 [math.NT], 2014.</div> <div class=sectline>Paul Barry, <a href="http://arxiv.org/abs/1205.2565">On sequences with {-1, 0, 1} Hankel transforms</a>, arXiv preprint arXiv:1205.2565 [math.CO], 2012. - From <a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</a>, Oct 18 2012</div> <div class=sectline>Burkard Polster (Mathologer), <a href="https://www.youtube.com/watch?v=iJ8pnCO0nTY">The hardest "What comes next?" (Euler's pentagonal formula)</a>, Youtube video, Oct 17 2020.</div> <div class=sectline>S. Cooper and M. D. Hirschhorn, <a href="http://dx.doi.org/10.1016/S0012-365X(03)00079-7">Results of Hurwitz type for three squares</a>, Discrete Math., Vol. 274, No. 1-3 (2004), pp. 9-24. See P(q).</div> <div class=sectline>Stephen Eberhart, <a href="/A001318/a001318_2.pdf">Letter to N. J. A. Sloane, Jan 19 1978</a>.</div> <div class=sectline>John Elias, <a href="/A001318/a001318.png">Illustration of Initial Terms: Generalized Penthexagrams</a>.</div> <div class=sectline>John Elias, <a href="/A001318/a001318_1.png">Illustration: Star Number Fractal</a>.</div> <div class=sectline>John Elias, <a href="/A001318/a001318_2.png">Illustration: Generalized Pentagonals In Generalized-Octagonal-Hexagrams</a>.</div> <div class=sectline>Andreas Enge, William Hart, and Fredrik Johansson, <a href="http://arxiv.org/abs/1608.06810">Short addition sequences for theta functions</a>, arXiv:1608.06810 [math.NT], 2016.</div> <div class=sectline>Leonhard Euler, <a href="https://scholarlycommons.pacific.edu/euler-works/175/">D茅couverte d'une loi tout extraordinaire des nombres par rapport 脿 la somme de leurs diviseurs</a>, Opera Omnia, Series I, Vol. 2 (1751), pp. 241-253.</div> <div class=sectline>Leonhard Euler, <a href="https://arxiv.org/abs/math/0505373">On the remarkable properties of the pentagonal numbers</a>, arXiv:math/0505373 [math.HO], 2005.</div> <div class=sectline>Leonhard Euler, <a href="http://math.dartmouth.edu/~euler/pages/E542.html">De mirabilibus proprietatibus numerorum pentagonalium</a>, par. 2</div> <div class=sectline>Leonhard Euler, <a href="http://math.dartmouth.edu/~euler/pages/E243.html">Observatio de summis divisorum</a> p. 8.</div> <div class=sectline>Leonhard Euler, <a href="https://arxiv.org/abs/math/0411587">An observation on the sums of divisors</a>, p. 8, arXiv:math/0411587 [math.HO], 2004.</div> <div class=sectline>Alex Fink, Richard K. Guy, and Mark Krusemeyer, <a href="https://cdm.ucalgary.ca/article/view/61940/46659">Partitions with parts occurring at most thrice</a>, Contrib. Discr. Math., Vol. 3, No. 2 (2008), pp. 76-114.</div> <div class=sectline>Silvia Heubach and Toufik Mansour, <a href="https://arxiv.org/abs/math/0310197">Counting rises, levels and drops in compositions</a>, arXiv:math/0310197 [math.CO], 2003.</div> <div class=sectline>Alfred Hoehn, <a href="/A001318/a001318.pdf">Illustration of initial terms</a>.</div> <div class=sectline>Barbara H. Margolius, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/MARGOLIUS/inversions.html">Permutations with inversions</a>, J. Integ. Seq., Vol. 4 (2001), Article 01.2.4.</div> <div class=sectline>Johannes W. Meijer, Euler's Ship on the Pentagonal Sea, <a href="/A001318/a001318_1.pdf">pdf</a> and <a href="/A001318/a001318.jpg">jpg</a>.</div> <div class=sectline>Johannes W. Meijer and Manuel Nepveu, <a href="http://ucbconocimiento.cba.ucb.edu.bo/index.php/RAN/article/download/485/427">Euler's ship on the Pentagonal Sea</a>, Acta Nova, Vol. 4, No. 1 (December 2008), pp. 176-187.</div> <div class=sectline>Mircea Merca, <a href="https://www.researchgate.net/publication/312324402">The Lambert series factorization theorem</a>, The Ramanujan Journal, January 2017; DOI: 10.1007/s11139-016-9856-3.</div> <div class=sectline>Mircea Merca and Maxie D. Schmidt, <a href="https://arxiv.org/abs/1706.02359">New Factor Pairs for Factorizations of Lambert Series Generating Functions</a>, arXiv:1706.02359 [math.CO], 2017. See Remark 2.2.</div> <div class=sectline>Mircea Merca, <a href="https://doi.org/10.1007/s40590-024-00652-1">Euler鈥檚 partition function in terms of 2-adic valuation</a>, Bol. Soc. Mat. Mex. 30, 76 (2024). See p. 3.</div> <div class=sectline>Ivan Niven, <a href="http://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/IvanNiven.pdf">Formal power series</a>, Amer. Math. Monthly, Vol. 76, No. 8 (1969), pp. 871-889.</div> <div class=sectline>Vladimir Pletser, <a href="http://arxiv.org/abs/1409.7969">Congruence conditions on the number of terms in sums of consecutive squared integers equal to squared integers</a>, arXiv:1409.7969 [math.NT], 2014.</div> <div class=sectline>Vladimir Pletser, <a href="http://arxiv.org/abs/1409.7972">Finding all squared integers expressible as the sum of consecutive squared integers using generalized Pell equation solutions with Chebyshev polynomials</a>, arXiv preprint arXiv:1409.7972 [math.NT], 2014.</div> <div class=sectline>Simon Plouffe, <a href="https://arxiv.org/abs/0911.4975">Approximations de s茅ries g茅n茅ratrices et quelques conjectures</a>, Dissertation, Universit茅 du Qu茅bec 脿 Montr茅al, 1992, arXiv:0911.4975 [math.NT], 2009.</div> <div class=sectline>Simon Plouffe, <a href="/A000051/a000051_2.pdf">1031 Generating Functions</a>, Appendix to Thesis, Montreal, 1992.</div> <div class=sectline>S. Realis, <a href="http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN598948236_0004&DMDID=DMDLOG_0076&IDDOC=630831">Question 271</a>, Nouv. Corresp. Math., 4 (1878) 27-29.</div> <div class=sectline>Steven J. Schlicker, <a href="http://www.jstor.org/stable/10.4169/math.mag.84.5.339">Numbers Simultaneously Polygonal and Centered Polygonal</a>, Mathematics Magazine, Vol. 84, No. 5 (December 2011), pp. 339-350.</div> <div class=sectline>Andr茅 Weil, <a href="http://dx.doi.org/10.5169/seals-46896">Two lectures on number theory, past and present</a>, L'Enseign. Math., Vol. XX (1974), pp. 87-110; Oeuvres III, pp. 279-302.</div> <div class=sectline>M. Wohlgemuth, <a href="http://matheplanet.com/default3.html?article=277">Pentagon, Kartenhaus und Summenzerlegung</a>.</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal numbers</a>, <a href="http://mathworld.wolfram.com/PartitionFunctionP.html">Partition Function P</a>.</div> <div class=sectline>Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumberTheorem.html">Pentagonal Number Theorem</a>.</div> <div class=sectline>Wikipedia, <a href="http://www.wikipedia.org/wiki/Pentagonal number theorem">Pentagonal number theorem</a>.</div> <div class=sectline>Keke Zhang, <a href="https://arxiv.org/abs/2011.09593">Generalized Catalan numbers</a>, arXiv:2011.09593 [math.CO], 2020.</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>Euler: Product_{n>=1} (1 - x^n) = Sum_{n=-oo..oo} (-1)^n*x^(n*(3*n - 1)/2).</div> <div class=sectline><a href="/A080995" title="Characteristic function of generalized pentagonal numbers A001318.">A080995</a>(a(n)) = 1: complement of <a href="/A090864" title="Complement of generalized pentagonal numbers (A001318).">A090864</a>; <a href="/A000009" title="Expansion of Product_{m >= 1} (1 + x^m); number of partitions of n into distinct parts; number of partitions of n into odd p...">A000009</a>(a(n)) = <a href="/A051044" title="Odd values of the PartitionsQ function A000009.">A051044</a>(n). - <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Apr 22 2006</div> <div class=sectline>Euler transform of length-3 sequence [2, 2, -1]. - <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Mar 24 2011</div> <div class=sectline>a(-1 - n) = a(n) for all n in Z. a(2*n) = <a href="/A005449" title="Second pentagonal numbers: a(n) = n*(3*n + 1)/2.">A005449</a>(n). a(2*n - 1) = <a href="/A000326" title="Pentagonal numbers: a(n) = n*(3*n-1)/2.">A000326</a>(n). - <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Mar 24 2011. [The extension of the recurrence to negative indices satisfies the signature (1,2,-2,-1,1), but not the definition of the sequence m*(3*m -1)/2, because there is no m such that a(-1) = 0. - <a href="/wiki/User:Klaus_Purath">Klaus Purath</a>, Jul 07 2021]</div> <div class=sectline>a(n) = 3 + 2*a(n-2) - a(n-4). - <a href="/wiki/User:Ant_King">Ant King</a>, Aug 23 2011</div> <div class=sectline>Product_{k>0} (1 - x^k) = Sum_{k>=0} (-1)^k * x^a(k). - <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Mar 24 2011</div> <div class=sectline>G.f.: x*(1 + x + x^2)/((1 + x)^2*(1 - x)^3).</div> <div class=sectline>a(n) = n*(n + 1)/6 when n runs through numbers == 0 or 2 mod 3. - <a href="/wiki/User:Barry_E._Williams">Barry E. Williams</a></div> <div class=sectline>a(n) = <a href="/A008805" title="Triangular numbers repeated.">A008805</a>(n-1) + <a href="/A008805" title="Triangular numbers repeated.">A008805</a>(n-2) + <a href="/A008805" title="Triangular numbers repeated.">A008805</a>(n-3), n > 2. - <a href="/wiki/User:Ralf_Stephan">Ralf Stephan</a>, Apr 26 2003</div> <div class=sectline>Sequence consists of the pentagonal numbers (<a href="/A000326" title="Pentagonal numbers: a(n) = n*(3*n-1)/2.">A000326</a>), followed by <a href="/A000326" title="Pentagonal numbers: a(n) = n*(3*n-1)/2.">A000326</a>(n) + n and then the next pentagonal number. - <a href="/wiki/User:Jon_Perry">Jon Perry</a>, Sep 11 2003</div> <div class=sectline>a(n) = (6*n^2 + 6*n + 1)/16 - (2*n + 1)*(-1)^n/16; a(n) = <a href="/A034828" title="a(n) = floor(n^2/4)*(n/2).">A034828</a>(n+1) - <a href="/A034828" title="a(n) = floor(n^2/4)*(n/2).">A034828</a>(n). - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, May 13 2005</div> <div class=sectline>a(n) = Sum_{k=1..floor((n+1)/2)} (n - k + 1). - <a href="/wiki/User:Paul_Barry">Paul Barry</a>, Sep 07 2005</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="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a>(floor(n/2)). - <a href="/wiki/User:Pierre_CAMI">Pierre CAMI</a>, Dec 09 2007</div> <div class=sectline>If n even a(n) = a(n-1) + n/2 and if n odd a(n) = a(n-1) + n, n >= 2. - <a href="/wiki/User:Pierre_CAMI">Pierre CAMI</a>, Dec 09 2007</div> <div class=sectline>a(n)-a(n-1) = <a href="/A026741" title="a(n) = n if n odd, n/2 if n even.">A026741</a>(n) and it follows that the difference between consecutive terms is equal to n if n is odd and to n/2 if n is even. Hence this is a self-generating sequence that can be simply constructed from knowledge of the first term alone. - <a href="/wiki/User:Ant_King">Ant King</a>, Sep 26 2011</div> <div class=sectline>a(n) = (1/2)*ceiling(n/2)*ceiling((3*n + 1)/2). - <a href="/wiki/User:Mircea_Merca">Mircea Merca</a>, Jul 13 2012</div> <div class=sectline>a(n) = (<a href="/A008794" title="Squares repeated; a(n) = floor(n/2)^2.">A008794</a>(n+1) + <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="/A002378" title="Oblong (or promic, pronic, or heteromecic) numbers: a(n) = n*(n+1).">A002378</a>(n) - <a href="/A085787" title="Generalized heptagonal numbers: m*(5*m - 3)/2, m = 0, +-1, +-2 +-3, ...">A085787</a>(n). - <a href="/wiki/User:Omar_E._Pol">Omar E. Pol</a>, Jan 12 2013</div> <div class=sectline>a(n) = floor((n + 1)/2)*((n + 1) - (1/2)*floor((n + 1)/2) - 1/2). - <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Jan 26 2013</div> <div class=sectline>From <a href="/wiki/User:Oskar_Wieland">Oskar Wieland</a>, Apr 10 2013: (Start)</div> <div class=sectline>a(n) = a(n+1) - <a href="/A026741" title="a(n) = n if n odd, n/2 if n even.">A026741</a>(n),</div> <div class=sectline>a(n) = a(n+2) - <a href="/A001651" title="Numbers not divisible by 3.">A001651</a>(n),</div> <div class=sectline>a(n) = a(n+3) - <a href="/A184418" title="Convolution square of A040001.">A184418</a>(n),</div> <div class=sectline>a(n) = a(n+4) - <a href="/A007310" title="Numbers congruent to 1 or 5 mod 6.">A007310</a>(n),</div> <div class=sectline>a(n) = a(n+6) - <a href="/A001651" title="Numbers not divisible by 3.">A001651</a>(n)*3 = a(n+6) - <a href="/A016051" title="Numbers of the form 9*k+3 or 9*k+6.">A016051</a>(n),</div> <div class=sectline>a(n) = a(n+8) - <a href="/A007310" title="Numbers congruent to 1 or 5 mod 6.">A007310</a>(n)*2 = a(n+8) - <a href="/A091999" title="Numbers that are congruent to {2, 10} mod 12.">A091999</a>(n),</div> <div class=sectline>a(n) = a(n+10)- <a href="/A001651" title="Numbers not divisible by 3.">A001651</a>(n)*5 = a(n+10)- <a href="/A072703" title="Indices of Fibonacci numbers whose last digit is 5.">A072703</a>(n),</div> <div class=sectline>a(n) = a(n+12)- <a href="/A007310" title="Numbers congruent to 1 or 5 mod 6.">A007310</a>(n)*3,</div> <div class=sectline>a(n) = a(n+14)- <a href="/A001651" title="Numbers not divisible by 3.">A001651</a>(n)*7. (End)</div> <div class=sectline>a(n) = (<a href="/A007310" title="Numbers congruent to 1 or 5 mod 6.">A007310</a>(n+1)^2 - 1)/24. - <a href="/wiki/User:Richard_R._Forberg">Richard R. Forberg</a>, May 27 2013; corrected by <a href="/wiki/User:Zak_Seidov">Zak Seidov</a>, Mar 14 2015; further corrected by <a href="/wiki/User:Jianing_Song">Jianing Song</a>, Oct 24 2018</div> <div class=sectline>a(n) = Sum_{i = ceiling((n+1)/2)..n} i. - <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Jun 08 2013</div> <div class=sectline>G.f.: x*G(0), where G(k) = 1 + x*(3*k + 4)/(3*k + 2 - x*(3*k + 2)*(3*k^2 + 11*k + 10)/(x*(3*k^2 + 11*k + 10) + (k + 1)*(3*k + 4)/G(k+1))); (continued fraction). - <a href="/wiki/User:Sergei_N._Gladkovskii">Sergei N. Gladkovskii</a>, Jun 16 2013</div> <div class=sectline>Sum_{n>=1} 1/a(n) = 6 - 2*Pi/sqrt(3). - <a href="/wiki/User:Vaclav_Kotesovec">Vaclav Kotesovec</a>, Oct 05 2016</div> <div class=sectline>a(n) = Sum_{i=1..n} numerator(i/2) = Sum_{i=1..n} denominator(2/i). - <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Feb 26 2017</div> <div class=sectline>a(n) = <a href="/A000292" title="Tetrahedral (or triangular pyramidal) numbers: a(n) = C(n+2,3) = n*(n+1)*(n+2)/6.">A000292</a>(<a href="/A001651" title="Numbers not divisible by 3.">A001651</a>(n))/<a href="/A001651" title="Numbers not divisible by 3.">A001651</a>(n), for n>0. - <a href="/wiki/User:Ivan_N._Ianakiev">Ivan N. Ianakiev</a>, May 08 2018</div> <div class=sectline>a(n) = ((-5 + (-1)^n - 6n)*(-1 + (-1)^n - 6n))/96. - <a href="/wiki/User:Jos茅_de_Jes煤s_Camacho_Medina">Jos茅 de Jes煤s Camacho Medina</a>, Jun 12 2018</div> <div class=sectline>a(n) = Sum_{k=1..n} k/gcd(k,2). - <a href="/wiki/User:Pedro_Caceres">Pedro Caceres</a>, Apr 23 2019</div> <div class=sectline>Quadrisection. For r = 0,1,2,3: a(r + 4*k) = 6*k^2 + sqrt(24*a(r) + 1)*k + a(r), for k >= 1, with inputs (k = 0) {0,1,2,5}. These are the sequences <a href="/A049453" title="Second pentagonal numbers with even index: a(n) = n*(6*n+1).">A049453</a>(k), <a href="/A033570" title="Pentagonal numbers with odd index: a(n) = (2*n+1)*(3*n+1).">A033570</a>(k), <a href="/A033568" title="Second pentagonal numbers with odd index: a(n) = (2*n-1)*(3*n-1).">A033568</a>(k+1), <a href="/A049452" title="Pentagonal numbers with even index.">A049452</a>(k+1), for k >= 0, respectively. - <a href="/wiki/User:Wolfdieter_Lang">Wolfdieter Lang</a>, Feb 12 2021</div> <div class=sectline>a(n) = a(n-4) + sqrt(24*a(n-2) + 1), n >= 4. - <a href="/wiki/User:Klaus_Purath">Klaus Purath</a>, Jul 07 2021</div> <div class=sectline>Sum_{n>=1} (-1)^(n+1)/a(n) = 6*(log(3)-1). - <a href="/wiki/User:Amiram_Eldar">Amiram Eldar</a>, Feb 28 2022</div> <div class=sectline>a(n) = <a href="/A002620" title="Quarter-squares: a(n) = floor(n/2)*ceiling(n/2). Equivalently, a(n) = floor(n^2/4).">A002620</a>(n) + <a href="/A008805" title="Triangular numbers repeated.">A008805</a>(n-1). <a href="/wiki/User:Gary_W._Adamson">Gary W. Adamson</a>, Dec 10 2022</div> <div class=sectline>E.g.f.: (x*(7 + 3*x)*cosh(x) + (1 + 5*x + 3*x^2)*sinh(x))/8. - <a href="/wiki/User:Stefano_Spezia">Stefano Spezia</a>, Aug 01 2024</div> </div> </div> <div class=section> <div class=sectname>EXAMPLE</div> <div class=sectbody> <div class=sectline>G.f. = x + 2*x^2 + 5*x^3 + 7*x^4 + 12*x^5 + 15*x^6 + 22*x^7 + 26*x^8 + 35*x^9 + ...</div> </div> </div> <div class=section> <div class=sectname>MAPLE</div> <div class=sectbody> <div class=sectline><a href="/A001318" title="Generalized pentagonal numbers: m*(3*m - 1)/2, m = 0, +-1, +-2, +-3, ....">A001318</a> := -(1+z+z**2)/(z+1)**2/(z-1)**3; # <a href="/wiki/User:Simon_Plouffe">Simon Plouffe</a> in his 1992 dissertation; gives sequence without initial zero</div> <div class=sectline><a href="/A001318" title="Generalized pentagonal numbers: m*(3*m - 1)/2, m = 0, +-1, +-2, +-3, ....">A001318</a> := proc(n) (6*n^2+6*n+1)/16-(2*n+1)*(-1)^n/16 ; end proc: # <a href="/wiki/User:R._J._Mathar">R. J. Mathar</a>, Mar 27 2011</div> </div> </div> <div class=section> <div class=sectname>MATHEMATICA</div> <div class=sectbody> <div class=sectline>Table[n*(n+1)/6, {n, Select[Range[0, 100], Mod[#, 3] != 1 &]}]</div> <div class=sectline>Select[Accumulate[Range[0, 200]]/3, IntegerQ] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Oct 12 2014 *)</div> <div class=sectline>CoefficientList[Series[x (1 + x + x^2) / ((1 + x)^2 (1 - x)^3), {x, 0, 70}], x] (* <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Nov 04 2014 *)</div> <div class=sectline>LinearRecurrence[{1, 2, -2, -1, 1}, {0, 1, 2, 5, 7}, 70] (* <a href="/wiki/User:Harvey_P._Dale">Harvey P. Dale</a>, Jun 05 2017 *)</div> <div class=sectline>a[ n_] := With[{m = Quotient[n + 1, 2]}, m (3 m + (-1)^n) / 2]; (* <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Jun 02 2018 *)</div> </div> </div> <div class=section> <div class=sectname>PROG</div> <div class=sectbody> <div class=sectline>(PARI) {a(n) = (3*n^2 + 2*n + (n%2) * (2*n + 1)) / 8}; /* <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Mar 24 2011 */</div> <div class=sectline>(PARI) {a(n) = if( n<0, n = -1-n); polcoeff( x * (1 - x^3) / ((1 - x) * (1-x^2))^2 + x * O(x^n), n)}; /* <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Mar 24 2011 */</div> <div class=sectline>(PARI) {a(n) = my(m = (n+1) \ 2); m * (3*m + (-1)^n) / 2}; /* <a href="/wiki/User:Michael_Somos">Michael Somos</a>, Jun 02 2018 */</div> <div class=sectline>(Sage)</div> <div class=sectline>@CachedFunction</div> <div class=sectline>def <a href="/A001318" title="Generalized pentagonal numbers: m*(3*m - 1)/2, m = 0, +-1, +-2, +-3, ....">A001318</a>(n):</div> <div class=sectline> if n == 0 : return 0</div> <div class=sectline> inc = n//2 if is_even(n) else n</div> <div class=sectline> return inc + <a href="/A001318" title="Generalized pentagonal numbers: m*(3*m - 1)/2, m = 0, +-1, +-2, +-3, ....">A001318</a>(n-1)</div> <div class=sectline>[<a href="/A001318" title="Generalized pentagonal numbers: m*(3*m - 1)/2, m = 0, +-1, +-2, +-3, ....">A001318</a>(n) for n in (0..59)] # <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, Oct 13 2012</div> <div class=sectline>(Magma) [(6*n^2 + 6*n + 1 - (2*n + 1)*(-1)^n)/16 : n in [0..50]]; // <a href="/wiki/User:Wesley_Ivan_Hurt">Wesley Ivan Hurt</a>, Nov 03 2014</div> <div class=sectline>(Magma) [(3*n^2 + 2*n + (n mod 2) * (2*n + 1)) div 8: n in [0..70]]; // <a href="/wiki/User:Vincenzo_Librandi">Vincenzo Librandi</a>, Nov 04 2014</div> <div class=sectline>(Haskell)</div> <div class=sectline>a001318 n = a001318_list !! n</div> <div class=sectline>a001318_list = scanl1 (+) a026741_list -- <a href="/wiki/User:Reinhard_Zumkeller">Reinhard Zumkeller</a>, Nov 15 2015</div> <div class=sectline>(GAP) a:=[0, 1, 2, 5];; for n in [5..60] do a[n]:=2*a[n-2]-a[n-4]+3; od; a; # <a href="/wiki/User:Muniru_A_Asiru">Muniru A Asiru</a>, Aug 16 2018</div> <div class=sectline>(Python)</div> <div class=sectline>def a(n):</div> <div class=sectline> p = n % 2</div> <div class=sectline> return (n + p)*(3*n + 2 - p) >> 3</div> <div class=sectline>print([a(n) for n in range(60)]) # <a href="/wiki/User:Peter_Luschny">Peter Luschny</a>, Jul 15 2022</div> <div class=sectline>(Python)</div> <div class=sectline>def <a href="/A001318" title="Generalized pentagonal numbers: m*(3*m - 1)/2, m = 0, +-1, +-2, +-3, ....">A001318</a>(n): return n*(n+1)-(m:=n>>1)*(m+1)>>1 # <a href="/wiki/User:Chai_Wah_Wu">Chai Wah Wu</a>, Nov 23 2024</div> </div> </div> <div class=section> <div class=sectname>CROSSREFS</div> <div class=sectbody> <div class=sectline>Cf. <a href="/A080995" title="Characteristic function of generalized pentagonal numbers A001318.">A080995</a> (characteristic function), <a href="/A026741" title="a(n) = n if n odd, n/2 if n even.">A026741</a> (first differences), <a href="/A034828" title="a(n) = floor(n^2/4)*(n/2).">A034828</a> (partial sums), <a href="/A165211" title="Period 8: repeat [0,1,0,1,1,0,1,0].">A165211</a> (mod 2).</div> <div class=sectline>Cf. <a href="/A000326" title="Pentagonal numbers: a(n) = n*(3*n-1)/2.">A000326</a> (pentagonal numbers), <a href="/A005449" title="Second pentagonal numbers: a(n) = n*(3*n + 1)/2.">A005449</a> (second pentagonal numbers), <a href="/A000217" title="Triangular numbers: a(n) = binomial(n+1,2) = n*(n+1)/2 = 0 + 1 + 2 + ... + n.">A000217</a> (triangular numbers).</div> <div class=sectline>Indices of nonzero terms of <a href="/A010815" title="From Euler's Pentagonal Theorem: coefficient of q^n in Product_{m>=1} (1 - q^m).">A010815</a>, i.e., the (zero-based) indices of 1-bits of the infinite binary word to which the terms of <a href="/A068052" title="Start from 1, shift one left and sum mod 2 (bitwise-XOR) to get 3 (11 in binary), then shift two steps left and XOR to get 1...">A068052</a> converge.</div> <div class=sectline>Union of <a href="/A036498" title="Numbers of the form m*(6*m-1) and m*(6*m+1), where m is an integer.">A036498</a> and <a href="/A036499" title="Numbers of the form k*(k+1)/6 for k = 2 or 3 modulo 6.">A036499</a>.</div> <div class=sectline>Cf. <a href="/A153384" title="Numbers n such that 24*n+1 is not prime.">A153384</a>, <a href="/A168258" title="Triangle read by rows, A101688 * A000012 as infinite lower triangular matrices.">A168258</a>, <a href="/A101688" title="Once 1, once 0, repeat, twice 1, twice 0, repeat, thrice 1, thrice 0, ... and so on.">A101688</a>, <a href="/A174739" title="Triangle read by rows, a partition number generator; A145006 * the diagonalized variant of A000041, (A174712).">A174739</a>, <a href="/A175005" title="Expansion of x/(1 - 4*x + 3*x^2 - 2*x^3).">A175005</a>.</div> <div class=sectline>Cf. <a href="/A074378" title="Even triangular numbers halved.">A074378</a>, <a href="/A057569" title="Numbers of the form k*(5*k+1)/2 or k*(5*k-1)/2.">A057569</a>, <a href="/A057570" title="Numbers of the form n*(7n+-1)/2.">A057570</a>, <a href="/A007310" title="Numbers congruent to 1 or 5 mod 6.">A007310</a>.</div> <div class=sectline>Sequences of generalized k-gonal numbers: this sequence (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), <a href="/A085787" title="Generalized heptagonal numbers: m*(5*m - 3)/2, m = 0, +-1, +-2 +-3, ...">A085787</a> (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>Column 1 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>Squares in APs: <a href="/A221671" title="Maximum number of squares in a non-constant arithmetic progression (AP) of length n.">A221671</a>, <a href="/A221672" title="Length of shortest non-constant arithmetic progression (AP) containing n squares.">A221672</a>.</div> <div class=sectline>Cf. <a href="/A054440" title="Number of ordered pairs of partitions of n with no common parts.">A054440</a>, <a href="/A260664" title="Number of ordered triples of partitions of n with no common parts.">A260664</a>, <a href="/A260672" title="Table read by rows: T(n,k) = n - A001318(k), k = 0 .. A193832(n)-1.">A260672</a>.</div> <div class=sectline>Quadrisection: <a href="/A049453" title="Second pentagonal numbers with even index: a(n) = n*(6*n+1).">A049453</a>(k), <a href="/A033570" title="Pentagonal numbers with odd index: a(n) = (2*n+1)*(3*n+1).">A033570</a>(k), <a href="/A033568" title="Second pentagonal numbers with odd index: a(n) = (2*n-1)*(3*n-1).">A033568</a>(k+1), <a href="/A049452" title="Pentagonal numbers with even index.">A049452</a>(k+1), k >= 0.</div> <div class=sectline>Cf. <a href="/A002620" title="Quarter-squares: a(n) = floor(n/2)*ceiling(n/2). Equivalently, a(n) = floor(n^2/4).">A002620</a>.</div> <div class=sectline>Sequence in context: <a href="/A129232" title="a(n) = floor(n*r) + floor((n-2)*r) + floor((n-4)*r) + ... + floor(k*r), where r = 2^(1/2) and k = 0 if n is even, k = 1 if n...">A129232</a> <a href="/A088822" title="a(n) is the sum of largest prime factors of numbers from 1 to n.">A088822</a> <a href="/A080182" title="a(1) = 1, a(n+1) = a(n) + gpf(Sum_{i=1..n} a(i)), where gpf=A006530 (greatest prime factor).">A080182</a> * <a href="/A024702" title="a(n) = (prime(n)^2 - 1)/24.">A024702</a> <a href="/A343944" title="Total number of parts in all partitions of n into powers of 2: p1 <= p2 <= ... <= p_k such that p_i <= 1 + Sum_{j=1..i-1} p_j.">A343944</a> <a href="/A226084" title="Number of partitions of n with Cookie Monster number 2.">A226084</a></div> <div class=sectline>Adjacent sequences: <a href="/A001315" title="a(n) = Sum_{k=0..n} 2^binomial(n,k).">A001315</a> <a href="/A001316" title="Gould's sequence: a(n) = Sum_{k=0..n} (binomial(n,k) mod 2); number of odd entries in row n of Pascal's triangle (A007318); ...">A001316</a> <a href="/A001317" title="Sierpi艅ski's triangle (Pascal's triangle mod 2) converted to decimal.">A001317</a> * <a href="/A001319" title="Number of (unordered) ways of making change for n cents using coins of 2, 5, 10, 20, 50 cents.">A001319</a> <a href="/A001320" title="Number of self-complementary Boolean functions of n variables: see Comments for precise definition.">A001320</a> <a href="/A001321" title="Number of equivalence classes of 3-valued Post functions of n variables under action of symmetric group S_n.">A001321</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>,<span title="an exceptionally nice sequence">nice</span>,<span title="edited within the last two weeks">changed</span></div> </div> </div> <div class=section> <div class=sectname>AUTHOR</div> <div class=sectbody> <div class=sectline><a href="/wiki/User:N._J._A._Sloane">N. J. A. Sloane</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 November 24 21:55 EST 2024. Contains 378084 sequences.</div> <div class=legal> <a href="/wiki/Legal_Documents">License Agreements, Terms of Use, Privacy Policy</a> </div> </div> </center> </div> </body> </html>