A380353 a(n) = (n^2 - n + 2) * (5*n^2 - 5*n + 2) / 4.

1, 12, 64, 217, 561, 1216, 2332, 4089, 6697, 10396, 15456, 22177, 30889, 41952, 55756, 72721, 93297, 117964, 147232, 181641, 221761, 268192, 321564, 382537, 451801, 530076, 618112, 716689, 826617, 948736, 1083916, 1233057, 1397089, 1576972, 1773696, 1988281, 2221777

Offset: 1

Author

Kelvin Voskuijl, Jan 22 2025

Keywords

• nonn
• easy

Comments

First differences of A072474 (sum of next n squares).

Links

• Kelvin Voskuijl, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Crossrefs

Cf. A072474 (partial sums), A051624, A000124.

Cf. A005448 (first difference of sum of next n natural numbers).

Programs

Mathematica

Table[((n^2 - n + 2)*(5*n^2 - 5*n + 2))/4, {n, 1, 40}]

PARI

a(n) = (n^2 - n + 2) * (5*n^2 - 5*n + 2) / 4

Formula

a(n) = A051624(A000124(n-1)).

G.f.: x*(1+3*x+x^2)*(1+4*x+x^2)/(1-x)^5. - Jinyuan Wang, Jan 23 2025

E.g.f.: exp(x)*(4 + 22*x^2 + 20*x^3 + 5*x^4)/4 - 1. - Stefano Spezia, Jan 28 2025