A195160 Generalized 11-gonal (or hendecagonal) numbers: m*(9*m - 7)/2 with m = 0, 1, -1, 2, -2, 3, -3, ...

0, 1, 8, 11, 25, 30, 51, 58, 86, 95, 130, 141, 183, 196, 245, 260, 316, 333, 396, 415, 485, 506, 583, 606, 690, 715, 806, 833, 931, 960, 1065, 1096, 1208, 1241, 1360, 1395, 1521, 1558, 1691, 1730, 1870, 1911, 2058, 2101, 2255, 2300, 2461, 2508, 2676

Offset: 0

Author

Omar E. Pol, Sep 10 2011

Keywords

• nonn
• easy

Comments

Exponents of q in the expansion of Product_{n >= 1} (1 - q^(9*n))*(1 + q^(9*n-1))*(1 + q^(9*n-8)) = 1 + q + q^8 + q^11 + q^25 + q^30 + .... - Peter Bala, Nov 21 2024

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (1,2,-2,-1,1).

Crossrefs

Partial sums of A195159.

Column 7 of A195152.

Cf. A316672.

Sequences of generalized k-gonal numbers: A001318 (k=5), A000217 (k=6), A085787 (k=7), A001082 (k=8), A118277 (k=9), A074377 (k=10), this sequence (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

I:=[0, 1, 8, 11, 25]; [n le 5 select I[n] else Self(n-1)+2*Self(n-2)-2*Self(n-3)-Self(n-4)+Self(n-5): n in [1..50]]; // Vincenzo Librandi, Apr 09 2013

Mathematica

CoefficientList[Series[x (1 + 7 x + x^2)/((1 + x)^2 (1 - x)^3), {x, 0, 60}], x] (* Vincenzo Librandi, Apr 09 2013 *)

PARI

a(n)=(18*n*(n+1)+5*(2*n+1)*(-1)^n-5)/16 \\ Charles R Greathouse IV, Sep 24 2015

Formula

From Bruno Berselli, Sep 14 2011: (Start)

G.f.: x*(1+7*x+x^2)/((1+x)^2*(1-x)^3).

a(n) = (18*n*(n+1)+5*(2*n+1)*(-1)^n-5)/16.

a(2n) = A062728(n), a(2n-1) = A051682(n). (End)

Sum_{n>=1} 1/a(n) = 18/49 + 2*Pi*cot(2*Pi/9)/7. - Vaclav Kotesovec, Oct 05 2016