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

Author

Jon Perry, Jul 23 2003

Keywords

• nonn
• easy

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

Examples

			From the first formula: a(5) = A000217(5) + A000217(2) = 15 + 3 = 18.
		

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).

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.

Programs

Haskell

a085787 n = a085787_list !! n
a085787_list = scanl (+) 0 a080512_list
-- Reinhard Zumkeller, Apr 06 2015

Magma

[5*n*(n+1)/8-1/16+(-1)^n*(2*n+1)/16: n in [0..60]]; // Vincenzo Librandi, Sep 11 2011

Mathematica

Select[Table[(n*(n+1)/2-1)/5,{n,500}],IntegerQ] (* Vladimir Joseph Stephan Orlovsky, Feb 06 2012 *)

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 */

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

Extensions

New name from T. D. Noe, Apr 21 2006

Formula in sequence name added by Omar E. Pol, May 28 2012