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

0,2

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).

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)

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 *)

Prog

(PARI) a(n)=(2*n-1)*(3*n-1) \\ Charles R Greathouse IV, Sep 24 2015
(Magma) [(2*n-1)*(3*n-1): n in [0..50]]; // G. C. Greubel, Oct 12 2019
(Sage) [(2*n-1)*(3*n-1) for n in range(50)] # G. C. Greubel, Oct 12 2019
(GAP) List([0..50], n-> (2*n-1)*(3*n-1)); # G. C. Greubel, Oct 12 2019

Crossrefs

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

Sequence in context: A180223 A070009 A070170 * A200156 A249997 A032016

Adjacent sequences: A033565 A033566 A033567 * A033569 A033570 A033571

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

More terms from Ray Chandler, Dec 08 2011

Status

approved