A033568 Second pentagonal numbers with odd index: a(n) = (2*n-1)*(3*n-1).

1, 2, 15, 40, 77, 126, 187, 260, 345, 442, 551, 672, 805, 950, 1107, 1276, 1457, 1650, 1855, 2072, 2301, 2542, 2795, 3060, 3337, 3626, 3927, 4240, 4565, 4902, 5251, 5612, 5985, 6370, 6767, 7176, 7597, 8030, 8475, 8932, 9401, 9882, 10375, 10880, 11397, 11926

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

Sequence found by reading the segment (1, 2) together with the line (one of the diagonal axes) from 2, in the direction 2, 15, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A001318, A005449, A033570, A049452, A049453, A051866.

Programs

GAP

List([0..50], n-> (2*n-1)*(3*n-1)); # G. C. Greubel, Oct 12 2019

Magma

[(2*n-1)*(3*n-1): n in [0..50]]; // G. C. Greubel, Oct 12 2019

Maple

seq((2*n-1)*(3*n-1), n=0..50); # G. C. Greubel, Oct 12 2019

Mathematica

Table[(2*n-1)*(3*n-1),{n,0,50}] (* Vladimir Joseph Stephan Orlovsky, Apr 28 2010 *)
LinearRecurrence[{3,-3,1},{1,2,15},50] (* Ray Chandler, Dec 08 2011 *)

PARI

a(n)=(2*n-1)*(3*n-1) \\ Charles R Greathouse IV, Sep 24 2015

Sage

[(2*n-1)*(3*n-1) for n in range(50)] # G. C. Greubel, Oct 12 2019

Formula

G.f.: (1-x+12*x^2)/(1-x)^3.

a(n) = a(n-1) + 12*n - 11 (with a(0)=1). - Vincenzo Librandi, Nov 17 2010

a(n) = 6*n^2 - 5*n + 1 = A051866(n) + 1. - Omar E. Pol, Jul 18 2012

E.g.f.: (1 + x + 6*x^2)*exp(x). - G. C. Greubel, Oct 12 2019

From Amiram Eldar, Feb 18 2022: (Start)

Sum_{n>=0} 1/a(n) = 1 + Pi/(2*sqrt(3)) + 2*log(2) - 3*log(3)/2.

Sum_{n>=0} (-1)^n/a(n) = 1 + (1/sqrt(3) - 1/2)*Pi - log(2). (End)

Extensions

More terms from Ray Chandler, Dec 08 2011