A129371 a(n) = Sum_{k=0..floor(n/2)} (n-k)^2.

0, 1, 5, 13, 29, 50, 86, 126, 190, 255, 355, 451, 595, 728, 924, 1100, 1356, 1581, 1905, 2185, 2585, 2926, 3410, 3818, 4394, 4875, 5551, 6111, 6895, 7540, 8440, 9176, 10200, 11033, 12189, 13125, 14421, 15466, 16910, 18070

Offset

0,3

Comments

Partial sums of A129370.

Partial row sums of A055461. - G. C. Greubel, Jan 31 2024

Links

G. C. Greubel, Table of n, a(n) for n = 0..5000

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

Formula

G.f.: x*(1+4*x+5*x^2+4*x^3)/((1-x)*(1-x^2)^3).

a(n) = Sum_{k = floor((n+1)/2)..n} k^2.

From R. J. Mathar, Apr 21 2010: (Start)

a(n) = a(n-1) +3*a(n-2) -3*a(n-3) -3*a(n-4) +3*a(n-5) +a(n-6) -a(n-7).

a(n) = 7*n^3/24 + 9*n^2/16 + 7*n/48 + (-1)^n*n*(n-1)/16. (End)

Mathematica

Accumulate[Table[n^2-(n-1)^2 (1-(-1)^n)/8, {n, 0, 50}]] (* Harvey P. Dale, Oct 22 2011 *)

Prog

(Magma) [n*(14*n^2+27*n+7)/48 +(-1)^n*Binomial(n, 2)/8: n in [0..60]]; // G. C. Greubel, Jan 31 2024
(SageMath) [n*(14*n^2+27*n+7)/48 +(-1)^n*binomial(n, 2)/8 for n in range(61)] # G. C. Greubel, Jan 31 2024

Crossrefs

Cf. A055461, A129370.

Sequence in context: A272750 A272801 A100438 * A212008 A194270 A194700

Adjacent sequences: A129368 A129369 A129370 * A129372 A129373 A129374

Keyword

easy,nonn

Author

Paul Barry, Apr 11 2007

Status

approved