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A085787 Generalized heptagonal numbers: n*(5*n-3)/2, n=0, +-1, +-2 +-3, ... 58
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

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(2n-1) = A000566(n).

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

EXAMPLE

a(5) = t(5)+t(2) = 15+8 = 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

Cf. A001318 [t(i)-t(floor(i/2))].

Column 3 of A195152.

Generalized k-gonal numbers, k>=5: A001318, A000217, this sequence, A001082, A118277, A074377, A195160, A195162, A195313, A195818.

Cf. A080512, A113429.

Sequence in context: A074136 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 6 05:15 EST 2016. Contains 278773 sequences.