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 A085787 Generalized heptagonal numbers: m*(5*m - 3)/2, m = 0, +-1, +-2 +-3, ... 84

%I

%S 0,1,4,7,13,18,27,34,46,55,70,81,99,112,133,148,172,189,216,235,265,

%T 286,319,342,378,403,442,469,511,540,585,616,664,697,748,783,837,874,

%U 931,970,1030,1071,1134,1177,1243,1288,1357,1404,1476,1525,1600,1651,1729

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

%C Zero together with the partial sums of A080512. - _Omar E. Pol_, Sep 10 2011

%C Second heptagonal numbers (A147875) and positive terms of A000566 interleaved. - _Omar E. Pol_, Aug 04 2012

%C 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

%C Characteristic function is A133100. - _Michael Somos_, Jan 30 2017

%C 40*a(n) + 9 is a square. - _Bruno Berselli_, Apr 18 2018

%C Numbers k such that the concatenation k225 is a square. - _Bruno Berselli_, Nov 07 2018

%D G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, p. 284.

%H Vincenzo Librandi, <a href="/A085787/b085787.txt">Table of n, a(n) for n = 0..10000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1)

%F a(n) = A000217(n) + A000217(floor(n/2)).

%F a(2*n-1) = A000566(n).

%F a(2*n) = A147875(n). - _Bruno Berselli_, Apr 18 2018

%F 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

%F a(n) = 5*n*(n + 1)/8 - 1/16 + (-1)^n*(2*n + 1)/16. - _R. J. Mathar_, Jun 29 2009

%F a(n) = (A000217(n) + A001082(n))/2 = (A001318(n) + A118277(n))/2. - _Omar E. Pol_, Jan 11 2013

%F a(n) = A002378(n) - A001318(n). - _Omar E. Pol_, Oct 23 2013

%F Sum_{n>=1} 1/a(n) = 10/9 + (2*sqrt(1 - 2/sqrt(5))*Pi)/3. - _Vaclav Kotesovec_, Oct 05 2016

%F E.g.f.: (x*(9 + 5*x)*exp(x) - (1 - 2*x)*sinh(x))/8. - _Franck Maminirina Ramaharo_, Nov 07 2018

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

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

%o (PARI) t(n)=n*(n+1)/2

%o for(i=0,40,print1(t(i)+t(floor(i/2)), ", "))

%o (PARI) {a(n) = (5*(-n\2)^2 - (-n\2)*3*(-1)^n) / 2}; /* _Michael Somos_, Oct 17 2006 */

%o (MAGMA) [5*n*(n+1)/8-1/16+(-1)^n*(2*n+1)/16: n in [0..60]]; // _Vincenzo Librandi_, Sep 11 2011

%o a085787 n = a085787_list !! n

%o a085787_list = scanl (+) 0 a080512_list

%o -- _Reinhard Zumkeller_, Apr 06 2015

%Y Column 3 of A195152.

%Y 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).

%Y Cf. A080512, A113429, A133100.

%K nonn,easy,changed

%O 0,3

%A _Jon Perry_, Jul 23 2003

%E New name from _T. D. Noe_, Apr 21 2006

%E Formula in sequence name added by _Omar E. Pol_, May 28 2012

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Last modified November 14 08:13 EST 2018. Contains 317174 sequences. (Running on oeis4.)