A085787 Generalized heptagonal numbers: m*(5*m - 3)/2, m = 0, +-1, +-2 +-3, ...

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

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

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). - Peter Bala, Dec 15 2024

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

Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, Cyclic permutations avoiding patterns in both one-line and cycle forms, arXiv:2312.05145 [math.CO], 2023. See p. 2.

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

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 (A001622). - Amiram Eldar, Feb 28 2022

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. A001622, A080512, A113429, A133100.

Sequence in context: A074136 A310824 A266811 * A111710 A191138 A075315

Adjacent sequences: A085784 A085785 A085786 * A085788 A085789 A085790

Keyword

nonn,easy,changed

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