A239325 a(n) = 6*n^2 + 8*n + 1.

1, 15, 41, 79, 129, 191, 265, 351, 449, 559, 681, 815, 961, 1119, 1289, 1471, 1665, 1871, 2089, 2319, 2561, 2815, 3081, 3359, 3649, 3951, 4265, 4591, 4929, 5279, 5641, 6015, 6401, 6799, 7209, 7631, 8065, 8511, 8969, 9439, 9921, 10415, 10921, 11439, 11969

Offset: 0

Author

Philippe Deléham, Mar 16 2014

Keywords

• nonn
• easy

Comments

Binomial transform of 1, 14, 12, 0, 0, 0 (0 continued).

Sum_{n>=0} 1/a(n) = (Psi(0,(4+sqrt(10))/6) - Psi(0,(4-sqrt(10))/6))/(2*sqrt(10)) = 1.14373625509612753878..., where Psi(n,k) is the n^th derivative of the digamma function. - Bruno Berselli, Mar 16 2014

Examples

			a(0) = 1*1 = 1;
a(1) = 1*1 + 14*1 = 15;
a(2) = 1*1 + 14*2 + 12*1 = 41;
a(3) = 1*1 + 14*3 + 12*3 = 79;
a(4) = 1*1 + 14*4 + 12*6 = 129; etc.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Crossrefs

Cf. A069133, A100536, A139267.

Programs

Mathematica

Table[6 n^2 + 8 n + 1, {n, 0, 44}] (* or *)
CoefficientList[Series[(1 + 12 x - x^2)/(1 - x)^3, {x, 0, 44}], x] (* Michael De Vlieger, Oct 04 2016 *)
LinearRecurrence[{3,-3,1},{1,15,41},50] (* Harvey P. Dale, May 11 2019 *)

PARI

a(n)=6*n^2+8*n+1 \\ Charles R Greathouse IV, Jun 17 2017

Formula

G.f.: (1 + 12*x - x^2)/(1-x)^3.

a(0) = 1, a(1) = 15, a(2) = 41; for n>2, a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3).

a(n) = C(n,0) + 14*C(n,1) + 12*C(n,2).

a(n) = (A069133(n+1) + A100536(n+1) - A000290(n))/2.

a(n) = A139267(n+1) - 1. - Yuriy Sibirmovsky, Oct 04 2016