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

Author

Paul Barry, Apr 11 2007

Keywords

• easy
• nonn

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).

Crossrefs

Cf. A055461, A129370.

Programs

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

Mathematica

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

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

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)