A049453 Second pentagonal numbers with even index: a(n) = n*(6*n+1).

0, 7, 26, 57, 100, 155, 222, 301, 392, 495, 610, 737, 876, 1027, 1190, 1365, 1552, 1751, 1962, 2185, 2420, 2667, 2926, 3197, 3480, 3775, 4082, 4401, 4732, 5075, 5430, 5797, 6176, 6567, 6970, 7385, 7812, 8251, 8702, 9165, 9640, 10127, 10626

Offset

0,2

Comments

Number of edges in the join of the complete tripartite graph of order 3n and the cycle graph of order n, K_n,n,n * C_n - Roberto E. Martinez II, Jan 07 2002

Sequence found by reading the line (one of the diagonal axes) from 0, in the direction 0, 7, ..., in the square spiral whose vertices are the generalized pentagonal numbers A001318. - Omar E. Pol, Sep 08 2011

First bisection of A036498. - Bruno Berselli, Nov 25 2012

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.: x*(7+5*x)/(1-x)^3.

a(n) = 12*n + a(n-1) - 5 with n>0, a(0)=0. - Vincenzo Librandi, Aug 06 2010

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3). - G. C. Greubel, Jun 07 2017

From Amiram Eldar, Feb 18 2022: (Start)

Sum_{n>=1} 1/a(n) = 6 - sqrt(3)*Pi/2 - 2*log(2) - 3*log(3)/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi + log(2) + sqrt(3)*log(2 + sqrt(3)) - 6. (End)

Maple

seq(binomial(6*n+1, 2)/3, n=0..42); # Zerinvary Lajos, Jan 21 2007

Mathematica

s=0; lst={s}; Do[s+=n++ +7; AppendTo[lst, s], {n, 0, 7!, 12}]; lst (* Vladimir Joseph Stephan Orlovsky, Nov 16 2008 *)
Table[n*(6*n + 1), {n, 0, 50}] (* G. C. Greubel, Jun 07 2017 *)

Prog

(PARI) x='x+O('x^50); concat([0], Vec(x*(7+5*x)/(1-x)^3)) \\ G. C. Greubel, Jun 07 2017

Crossrefs

Cf. A001318, A005449, A033568, A033570, A036498, A049452, A185019, A194454.

Cf. A254963 (comment).

Sequence in context: A274268 A059376 A206481 * A231888 A211645 A171340

Adjacent sequences: A049450 A049451 A049452 * A049454 A049455 A049456

Keyword

nonn,easy

Author

Joe Keane (jgk(AT)jgk.org)

Status

approved