A027903 a(n) = n*(n + 1)*(3*n + 1).

0, 8, 42, 120, 260, 480, 798, 1232, 1800, 2520, 3410, 4488, 5772, 7280, 9030, 11040, 13328, 15912, 18810, 22040, 25620, 29568, 33902, 38640, 43800, 49400, 55458, 61992, 69020, 76560, 84630, 93248, 102432, 112200, 122570, 133560, 145188, 157472, 170430

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• changed

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Crossrefs

Cf. A000567, A001477, A002378, A016742, A016777, A049451, A117642.

Programs

Maple

A027903:=n->n*(n + 1)*(3*n + 1); seq(A027903(n), n=0..100); # Wesley Ivan Hurt, Dec 02 2013

Mathematica

Table[n (n + 1) (3*n + 1), {n, 0, 100}] (* Wesley Ivan Hurt, Dec 02 2013 *)

Formula

From Wesley Ivan Hurt, Dec 02 2013: (Start)

a(n) = n*(n + 1)*(3*n + 1).

a(n) = 3*n^3 + 4*n^2 + n.

a(n) = A002378(n) * A016777(n).

a(n) = A049451(n) * A001477(n+1).

a(n) = A001477(n) * A000567(n-1).

a(n) = A001477(n) * A001477(n+1) * A016777(n).

a(n) = A117642(n) + A016742(n) + A001477(n). (End)

From Amiram Eldar, Aug 15 2025: (Start)

Sum_{n>=1} 1/a(n) = 4 - sqrt(3)*Pi/4 - 9*log(3)/4.

Sum_{n>=1} (-1)^(n+1)/a(n) = sqrt(3)*Pi/2 + 2*log(2) - 4. (End)

From Elmo R. Oliveira, Aug 29 2025: (Start)

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

E.g.f.: x*(8 + 13*x + 3*x^2)*exp(x).

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