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 A085787 Generalized heptagonal numbers: m*(5*m - 3)/2, m = 0, +-1, +-2 +-3, ... 84
 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS Zero together with the partial sums of A080512. - Omar E. Pol, Sep 10 2011 Second heptagonal numbers (A147875) and positive terms of A000566 interleaved. - Omar E. Pol, Aug 04 2012 These numbers appear in a theta function identity. See the Hardy-Wright reference, Theorem 355 on p. 284. See the g.f. of A113429. - Wolfdieter Lang, Oct 28 2016 Characteristic function is A133100. - Michael Somos, Jan 30 2017 40*a(n) + 9 is a square. - Bruno Berselli, Apr 18 2018 Numbers k such that the concatenation k225 is a square. - Bruno Berselli, Nov 07 2018 REFERENCES G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, p. 284. LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..10000 Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1) FORMULA a(n) = A000217(n) + A000217(floor(n/2)). a(2*n-1) = A000566(n). a(2*n) = A147875(n). - Bruno Berselli, Apr 18 2018 G.f.: x * (1 + 3*x + x^2) / ((1 - x) * (1 - x^2)^2). a(n) = a(-1-n) for all n in Z. - Michael Somos, Oct 17 2006 a(n) = 5*n*(n + 1)/8 - 1/16 + (-1)^n*(2*n + 1)/16. - R. J. Mathar, Jun 29 2009 a(n) = (A000217(n) + A001082(n))/2 = (A001318(n) + A118277(n))/2. - Omar E. Pol, Jan 11 2013 a(n) = A002378(n) - A001318(n). - Omar E. Pol, Oct 23 2013 Sum_{n>=1} 1/a(n) = 10/9 + (2*sqrt(1 - 2/sqrt(5))*Pi)/3. - Vaclav Kotesovec, Oct 05 2016 E.g.f.: (x*(9 + 5*x)*exp(x) - (1 - 2*x)*sinh(x))/8. - Franck Maminirina Ramaharo, Nov 07 2018 EXAMPLE From the first formula: a(5) = A000217(5) + A000217(2) = 15 + 3 = 18. MATHEMATICA Select[Table[(n*(n+1)/2-1)/5, {n, 500}], IntegerQ] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2012 *) PROG (PARI) t(n)=n*(n+1)/2 for(i=0, 40, print1(t(i)+t(floor(i/2)), ", ")) (PARI) {a(n) = (5*(-n\2)^2 - (-n\2)*3*(-1)^n) / 2}; /* Michael Somos, Oct 17 2006 */ (MAGMA) [5*n*(n+1)/8-1/16+(-1)^n*(2*n+1)/16: n in [0..60]]; // Vincenzo Librandi, Sep 11 2011 (Haskell) a085787 n = a085787_list !! n a085787_list = scanl (+) 0 a080512_list -- Reinhard Zumkeller, Apr 06 2015 CROSSREFS Column 3 of A195152. Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), this sequence (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), A195160 (k=11), A195162 (k=12), A195313 (k=13), A195818 (k=14), A277082 (k=15), A274978 (k=16), A303305 (k=17), A274979 (k=18), A303813 (k=19), A218864 (k=20), A303298 (k=21), A303299 (k=22), A303303 (k=23), A303814 (k=24), A303304 (k=25), A316724 (k=26), A316725 (k=27), A303812 (k=28), A303815 (k=29), A316729 (k=30). Cf. A080512, A113429, A133100. Sequence in context: A310824 A266811 * A111710 A191138 A075315 A238327 Adjacent sequences:  A085784 A085785 A085786 * A085788 A085789 A085790 KEYWORD nonn,easy AUTHOR Jon Perry, Jul 23 2003 EXTENSIONS New name from T. D. Noe, Apr 21 2006 Formula in sequence name added by Omar E. Pol, May 28 2012 STATUS approved

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Last modified December 9 22:37 EST 2018. Contains 318032 sequences. (Running on oeis4.)