A267522 a(n) = 4*(n + 1)*(n + 2)*(4*n + 3)/3.

8, 56, 176, 400, 760, 1288, 2016, 2976, 4200, 5720, 7568, 9776, 12376, 15400, 18880, 22848, 27336, 32376, 38000, 44240, 51128, 58696, 66976, 76000, 85800, 96408, 107856, 120176, 133400, 147560, 162688, 178816, 195976, 214200, 233520, 253968, 275576, 298376, 322400

Offset: 0

Author

Ilya Gutkovskiy, Apr 09 2016

Keywords

• nonn
• easy

Comments

Partial sums of A152750.

Examples

			a(0) = (0 + 2)*(1 + 3) = 8;
a(1) = (0 + 2)*(1 + 3) + (2 + 4)*(3 + 5) = 56;
a(2) = (0 + 2)*(1 + 3) + (2 + 4)*(3 + 5) + (4 + 6)*(5 + 7) = 176;
a(3) = (0 + 2)*(1 + 3) + (2 + 4)*(3 + 5) + (4 + 6)*(5 + 7) + (6 + 8)*(7 + 9) = 400, etc
		

Links

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

Crossrefs

Cf. A002412, A152750, A268684.

Programs

Mathematica

Table[(4 (n + 1)) (n + 2) ((4 n + 3)/3), {n, 0, 38}]
LinearRecurrence[{4, -6, 4, -1}, {8, 56, 176, 400}, 39]

PARI

a(n) = 4*(n + 1)*(n + 2)*(4*n + 3)/3; \\ Michel Marcus, Apr 10 2016

PARI

x='x+O('x^99); Vec(8*(1+3*x)/(1-x)^4) \\ Altug Alkan, Apr 10 2016

Formula

G.f.: 8*(1 + 3*x)/(1 - x)^4.

E.g.f.: (4/3)*exp(x)*(6 + 36*x + 27*x^2 + 4*x^3).

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

a(n) = 4*A268684(n + 1).

Sum_{n>=0} 1/a(n) = -3*(2*Pi - 12*log(2) + 1)/20 = 0.15518712893...

a(n) = 8*A002412(n+1). - Yasser Arath Chavez Reyes, Feb 23 2024