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A085787 Generalized heptagonal numbers: m*(5*m - 3)/2, m = 0, +-1, +-2 +-3, ... 86
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
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
REFERENCES
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, p. 284.
LINKS
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.
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).
Sequence in context: A074136 A310824 A266811 * A111710 A191138 A075315
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 March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)