A185395 a(3n) = n^2, a(3n+1) = a(3n+2) = 3*n*(n+1)/2.

0, 0, 0, 1, 3, 3, 4, 9, 9, 9, 18, 18, 16, 30, 30, 25, 45, 45, 36, 63, 63, 49, 84, 84, 64, 108, 108, 81, 135, 135, 100, 165, 165, 121, 198, 198, 144, 234, 234, 169, 273, 273, 196, 315, 315, 225, 360, 360, 256

Offset: 0

Author

Philippe Deléham, Jan 21 2012

Keywords

• easy
• nonn

Comments

Expansion of ((x+x^2)/(1-x^3))^k with k = 3 ; for k = 1 see A011655, for k = 2 see A186731, for k = 4 see A185292.

Links

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

Crossrefs

Column k = 3 of triangle in A198295.

Cf. A000217, A000290, A011655, A045943, A072691, A186731, A185292.

Programs

Mathematica

LinearRecurrence[{0,0,3,0,0,-3,0,0,1},{0,0,0,1,3,3,4,9,9},50] (* Harvey P. Dale, Jan 23 2013 *)

PARI

x='x+O('x^50); concat([0, 0, 0], Vec((x*(1+x)/(1-x^3))^3)) \\ G. C. Greubel, Jun 29 2017

Formula

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

From Amiram Eldar, May 10 2025: (Start)

Sum_{n>=3} 1/a(n) = Pi^2/6 + 4/3.

Sum_{n>=3} (-1)^(n+1)/a(n) = Pi^2/12 (A072691). (End)