A152741 13 times triangular numbers.

0, 13, 39, 78, 130, 195, 273, 364, 468, 585, 715, 858, 1014, 1183, 1365, 1560, 1768, 1989, 2223, 2470, 2730, 3003, 3289, 3588, 3900, 4225, 4563, 4914, 5278, 5655, 6045, 6448, 6864, 7293, 7735, 8190, 8658, 9139, 9633, 10140, 10660, 11193, 11739, 12298, 12870

Offset: 0

Author

Omar E. Pol, Dec 12 2008

Keywords

• easy
• nonn

Comments

Sequence found by reading the line from 0, in the direction 0, 13,... and the same line from 0, in the direction 0, 39,..., in the square spiral whose vertices are the generalized 15-gonal numbers. - Omar E. Pol, Oct 03 2011

Sum of the numbers from 6n to 7n. - Wesley Ivan Hurt, Dec 22 2015

Links

• G. C. Greubel, Table of n, a(n) for n = 0..5000
• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A000217, A049598, A069126.

Programs

Magma

[13*n*(n+1)/2 : n in [0..60]]; // Wesley Ivan Hurt, Dec 22 2015

Maple

A152741:=n->13*n*(n+1)/2: seq(A152741(n), n=0..60); # Wesley Ivan Hurt, Dec 22 2015

Mathematica

Table[13*n*(n-1)/2, {n,100}] (* Vladimir Joseph Stephan Orlovsky, Jul 06 2011 *)
CoefficientList[Series[13 x/(1 - x)^3, {x, 0, 50}], x] (* Wesley Ivan Hurt, Dec 22 2015 *)

PARI

a(n)=13*n*(n+1)/2 \\ Charles R Greathouse IV, Jun 17 2017

Formula

a(n) = 13*n*(n+1)/2 = 13 * A000217(n).

a(n) = a(n-1)+13*n (with a(0)=0). - Vincenzo Librandi, Nov 26 2010

a(n) = A069126(n+1) - 1. - Omar E. Pol, Oct 03 2011

From Wesley Ivan Hurt, Dec 22 2015: (Start)

G.f.: 13*x/(1-x)^3.

a(n) = 3*a(n-1)-3*a(n-2)+a(n-3) for n>2.

a(n) = Sum_{i=6n..7n} i. (End)

E.g.f.: 13*x*(2+x)*exp(x)/2. - G. C. Greubel, Sep 01 2018

From Amiram Eldar, Feb 21 2023: (Start)

Sum_{n>=1} 1/a(n) = 2/13.

Sum_{n>=1} (-1)^(n+1)/a(n) = (4*log(2) - 2)/13.

Product_{n>=1} (1 - 1/a(n)) = -(13/(2*Pi))*cos(sqrt(21/13)*Pi/2).

Product_{n>=1} (1 + 1/a(n)) = (13/(2*Pi))*cos(sqrt(5/13)*Pi/2). (End)