A367965 a(n) = (1/8)*(4*n^2 + 6*n + (-1)^n*(2*n*(n + 1) - 1) + 1).

0, 1, 5, 4, 16, 9, 33, 16, 56, 25, 85, 36, 120, 49, 161, 64, 208, 81, 261, 100, 320, 121, 385, 144, 456, 169, 533, 196, 616, 225, 705, 256, 800, 289, 901, 324, 1008, 361, 1121, 400, 1240, 441, 1365, 484, 1496, 529, 1633, 576, 1776, 625, 1925, 676, 2080, 729, 2241, 784

Offset: 0

Author

Peter Luschny, Dec 07 2023

Keywords

• nonn
• easy

Links

• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A367964, A000290, A045944.

Programs

Maple

a := n -> (1/8)*(4*n^2 + 6*n + (-1)^n*(2*n*(n + 1) - 1) + 1):
seq(a(n), n = 0..55);

Mathematica

LinearRecurrence[{0,3,0,-3,0,1},{0,1,5,4,16,9},100] (* Paolo Xausa, Dec 07 2023 *)

Formula

a(n) = Sum_{k=0..n} (-1)^(n - k) * A367964(n, k).

a(2*n) = n*(3*n+2) = A045944(n).

a(2*n-1) = n^2 = A000290(n).

G.f.: x*(1 + 5*x + x^2 + x^3)/((1 - x)^3*(1 + x)^3). - Stefano Spezia, Dec 07 2023

Sum_{n>=1} 1/a(n) = Pi^2/6 + Pi/(4*sqrt(3)) - 3*(log(3)-1)/4. - Amiram Eldar, Dec 06 2024