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

%I A085787 #109 Dec 16 2024 08:26:49
%S A085787 0,1,4,7,13,18,27,34,46,55,70,81,99,112,133,148,172,189,216,235,265,
%T A085787 286,319,342,378,403,442,469,511,540,585,616,664,697,748,783,837,874,
%U A085787 931,970,1030,1071,1134,1177,1243,1288,1357,1404,1476,1525,1600,1651,1729
%N A085787 Generalized heptagonal numbers: m*(5*m - 3)/2, m = 0, +-1, +-2 +-3, ...
%C A085787 Zero together with the partial sums of A080512. - _Omar E. Pol_, Sep 10 2011
%C A085787 Second heptagonal numbers (A147875) and positive terms of A000566 interleaved. - _Omar E. Pol_, Aug 04 2012
%C A085787 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 A085787 Characteristic function is A133100. - _Michael Somos_, Jan 30 2017
%C A085787 40*a(n) + 9 is a square. - _Bruno Berselli_, Apr 18 2018
%C A085787 Numbers k such that the concatenation k225 is a square. - _Bruno Berselli_, Nov 07 2018
%C A085787 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
%D A085787 G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, p. 284.
%H A085787 Vincenzo Librandi, <a href="/A085787/b085787.txt">Table of n, a(n) for n = 0..10000</a>
%H A085787 Kassie Archer, Ethan Borsh, Jensen Bridges, Christina Graves, and Millie Jeske, <a href="https://arxiv.org/abs/2312.05145">Cyclic permutations avoiding patterns in both one-line and cycle forms</a>, arXiv:2312.05145 [math.CO], 2023. See p. 2.
%H A085787 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,2,-2,-1,1).
%F A085787 a(n) = A000217(n) + A000217(floor(n/2)).
%F A085787 a(2*n-1) = A000566(n).
%F A085787 a(2*n) = A147875(n). - _Bruno Berselli_, Apr 18 2018
%F A085787 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 A085787 a(n) = 5*n*(n + 1)/8 - 1/16 + (-1)^n*(2*n + 1)/16. - _R. J. Mathar_, Jun 29 2009
%F A085787 a(n) = (A000217(n) + A001082(n))/2 = (A001318(n) + A118277(n))/2. - _Omar E. Pol_, Jan 11 2013
%F A085787 a(n) = A002378(n) - A001318(n). - _Omar E. Pol_, Oct 23 2013
%F A085787 Sum_{n>=1} 1/a(n) = 10/9 + (2*sqrt(1 - 2/sqrt(5))*Pi)/3. - _Vaclav Kotesovec_, Oct 05 2016
%F A085787 E.g.f.: (x*(9 + 5*x)*exp(x) - (1 - 2*x)*sinh(x))/8. - _Franck Maminirina Ramaharo_, Nov 07 2018
%F A085787 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
%e A085787 From the first formula: a(5) = A000217(5) + A000217(2) = 15 + 3 = 18.
%t A085787 Select[Table[(n*(n+1)/2-1)/5,{n,500}],IntegerQ] (* _Vladimir Joseph Stephan Orlovsky_, Feb 06 2012 *)
%o A085787 (PARI) t(n)=n*(n+1)/2
%o A085787 for(i=0,40,print1(t(i)+t(floor(i/2)), ", "))
%o A085787 (PARI) {a(n) = (5*(-n\2)^2 - (-n\2)*3*(-1)^n) / 2}; /* _Michael Somos_, Oct 17 2006 */
%o A085787 (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 (Haskell)
%o A085787 a085787 n = a085787_list !! n
%o A085787 a085787_list = scanl (+) 0 a080512_list
%o A085787 -- _Reinhard Zumkeller_, Apr 06 2015
%Y A085787 Column 3 of A195152.
%Y A085787 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 A085787 Cf. A001622, A080512, A113429, A133100.
%K A085787 nonn,easy
%O A085787 0,3
%A A085787 _Jon Perry_, Jul 23 2003
%E A085787 New name from _T. D. Noe_, Apr 21 2006
%E A085787 Formula in sequence name added by _Omar E. Pol_, May 28 2012