A118277 Generalized 9-gonal (or enneagonal) numbers: m*(7*m - 5)/2 with m = 0, 1, -1, 2, -2, 3, -3, ...

0, 1, 6, 9, 19, 24, 39, 46, 66, 75, 100, 111, 141, 154, 189, 204, 244, 261, 306, 325, 375, 396, 451, 474, 534, 559, 624, 651, 721, 750, 825, 856, 936, 969, 1054, 1089, 1179, 1216, 1311, 1350, 1450, 1491, 1596, 1639, 1749, 1794, 1909, 1956, 2076, 2125, 2250

Offset: 0

Author

T. D. Noe, Apr 21 2006

Keywords

• nonn
• easy

Comments

Partial sums of A195140. - Omar E. Pol, Sep 13 2011

The characteristic function starts 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0 , ... and has the generating function f(x,x^6) in terms of Ramanujan's two-variable theta function. See A080995, A010054, A133100 etc. - Omar E. Pol, Jul 13 2012

Also A179986 and positive terms of A001106 interleaved. - Omar E. Pol, Aug 04 2012

Sequence provides all integers m such that 56*m + 25 is a square. - Bruno Berselli, Oct 07 2015

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).

Crossrefs

Cf. A001106 (9-gonal numbers).

Column 5 of A195152.

Cf. A195140.

Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), this sequence (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).

Programs

Magma

[7*n^2/8+7*n/8-3/16+3*(-1)^n*(1/16+n/8): n in [0..50]]; // Vincenzo Librandi, Oct 10 2011

Mathematica

n=9; Union[Table[i((n-2)i-(n-4))/2, {i,-30,30}]]
LinearRecurrence[{1,2,-2,-1,1},{0,1,6,9,19},60] (* Harvey P. Dale, Jun 08 2016 *)

PARI

a(n)=7*n*(n+1)/8-3/16+3*(-1)^n*(1+2*n)/16 \\ Charles R Greathouse IV, Jan 18 2012

Formula

a(n) = n*(7*n-5)/2 for positive and negative n.

a(n) = (1/16)*(14*n^2 + 14*n - 3 + 3*(-1)^n*(2*n + 1)). - R. J. Mathar, Oct 08 2011

G.f.: x*(1+5*x+x^2) / ( (1+x)^2*(1-x)^3 ). - R. J. Mathar, Oct 08 2011

Sum_{n>=1} 1/a(n) = 2*(7 + 5*Pi*tan(3*Pi/14))/25. - Vaclav Kotesovec, Oct 05 2016

E.g.f.: (1/16)*(3*(1 - 2*x)*exp(-x) + (-3 + 28*x + 14*x^2)*exp(x)). - G. C. Greubel, Aug 19 2017

Extensions

Extended Name by Omar E. Pol, Jul 28 2018