A225144 a(n) = Sum_{i=n..2*n} i^2*(-1)^i.

0, 3, 11, 18, 42, 45, 93, 84, 164, 135, 255, 198, 366, 273, 497, 360, 648, 459, 819, 570, 1010, 693, 1221, 828, 1452, 975, 1703, 1134, 1974, 1305, 2265, 1488, 2576, 1683, 2907, 1890, 3258, 2109, 3629, 2340, 4020, 2583, 4431, 2838, 4862, 3105, 5313, 3384

Offset: 0

Author

Bruno Berselli, Jun 06 2013

Keywords

• nonn
• easy

Comments

3 and 11 are the only primes in the sequence.

Examples

			a(6) = 6^2-7^2+8^2-9^2+10^2-11^2+12^2 = 93.
a(7) = -7^2+8^2-9^2+10^2-11^2+12^2-13^2+14^2 = 84.
		

Links

• Bruno Berselli, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (0,3,0,-3,0,1).

Crossrefs

Cf. A050409: sum(i^2, i=n..2n); A064455: sum(i*(-1)^i, i=n..2n); A065679: A000217(n)+(-1)^n*A000217(n-1); A089594: sum(i^2*(-1)^i, i=1..n).

Programs

Magma

[&+[i^2*(-1)^i: i in [n..2*n]]: n in [0..50]];

Mathematica

Table[Sum[i^2 (-1)^i, {i, n, 2 n}], {n, 0, 50}]

Formula

G.f.: x*(3+11*x+9*x^2+9*x^3)/(1-x^2)^3.

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

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

a(n) = A000217(2n) +(-1)^n*A000217(n-1) with A000217(-1)=0.

a(2n-1) = A094159(n) for n>0; a(2n) = A055437(n) for A055437(0)=0.