A391997 a(n) = Sum_{k=0..n} floor((k/2)^2)*n^2. Row sums of A391996.

0, 0, 4, 27, 112, 325, 792, 1666, 3200, 5670, 9500, 15125, 23184, 34307, 49392, 69300, 95232, 128316, 170100, 222015, 286000, 363825, 457864, 570262, 703872, 861250, 1045772, 1260441, 1509200, 1795535, 2124000, 2498600, 2924544, 3406392, 3950052, 4560675, 5244912

Offset

0,3

Links

Table of n, a(n) for n=0..36.

Index entries for linear recurrences with constant coefficients, signature (3,0,-8,6,6,-8,0,3,-1).

Formula

a(n) = [x^n] x^2*(9*x^4 + 21*x^3 + 31*x^2 + 15*x + 4) / ((x - 1)^6*(x + 1)^3).

a(n) ~ n^5/12. - Charles R Greathouse IV, May 31 2026

Maple

seq(add(floor((k/2)^2)*n^2, k = 0..n), n = 0..36);
gf := (9*x^4 + 21*x^3 + 31*x^2 + 15*x + 4)*x^2/((x - 1)^6*(x + 1)^3):
ser := series(gf, x, 40): seq(coeff(ser, x, n), n = 0..36);

Mathematica

LinearRecurrence[{3, 0, -8, 6, 6, -8, 0, 3, -1}, {0, 0, 4, 27, 112, 325, 792, 1666, 3200}, 40] (* Hugo Pfoertner, Jan 10 2026 *)

Prog

(PARI) a(n)=(2*n^5+3*n^4-2*n^3-n%2*3*n^2)/24 \\ Charles R Greathouse IV, May 31 2026

Crossrefs

Cf. A391997, A391996.

Sequence in context: A063262 A156223 A048102 * A171469 A267685 A190584

Adjacent sequences: A391994 A391995 A391996 * A391998 A391999 A392000

Keyword

nonn,easy,changed

Author

Peter Luschny, Jan 10 2026

Status

approved