A191745 a(n) = 12*n^3 + 9*n^2 + 2*n.

0, 23, 136, 411, 920, 1735, 2928, 4571, 6736, 9495, 12920, 17083, 22056, 27911, 34720, 42555, 51488, 61591, 72936, 85595, 99640, 115143, 132176, 150811, 171120, 193175, 217048, 242811, 270536, 300295, 332160, 366203, 402496, 441111, 482120, 525595, 571608

Offset: 0

Author

Adi Dani, Jun 14 2011

Keywords

• nonn
• easy
• changed

Comments

Number of partitions of 12*n+2 into 4 parts.

Examples

			a(1)=23: there are 23 partitions of 12*1+2=14 into 4 parts: [1,1,1,11], [1,1,2,10], [1,1,3,9], [1,1,4,8], [1,1,5,7], [1,1,6,6], [1,2,2,9], [1,2,3,8], [1,2,4,7], [1,2,5,6], [1,3,3,7], [1,3,4,6], [1,3,5,5], [1,4,4,5], [2,2,2,8], [2,2,3,7], [2,2,4,6], [2,2,5,5], [2,3,3,6], [2,3,4,5], [2,4,4,4], [3,3,3,5], [3,3,4,4].
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Programs

Magma

[12*n^3+9*n^2+2*n: n in [0..40]]; // Vincenzo Librandi, Jun 14 2011

Mathematica

Table[12n^3 + 9n^2 + 2n, {n, 0, 30}]
LinearRecurrence[{4,-6,4,-1},{0,23,136,411},40] (* Harvey P. Dale, Nov 05 2019 *)

PARI

a(n)=((12*n+9)*n+2)*n /* Charles R Greathouse IV, Jun 14 2011 */

Formula

From Elmo R. Oliveira, Aug 28 2025: (Start)

G.f.: x*(23 + 44*x + 5*x^2)/(x-1)^4.

E.g.f.: x*(23 + 45*x + 12*x^2)*exp(x).

a(n) = 4*a(n-1) - 6*a(n-2) + 4*a(n-3) - a(n-4). (End)