A059859 Sum of squares of first n quarter-squares (A002620).

0, 0, 1, 5, 21, 57, 138, 282, 538, 938, 1563, 2463, 3759, 5523, 7924, 11060, 15156, 20340, 26901, 35001, 45001, 57101, 71742, 89166, 109902, 134238, 162799, 195923, 234339, 278439, 329064, 386664, 452200, 526184, 609705, 703341

Offset

0,4

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

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

Formula

If n is even, a(n) = n*(n+2)*(2*n^3+n^2-2*n+4)/160; if n is odd, a(n) = (n^2-1)*(2*n^3+5*n^2+2*n-5)/160.

From R. J. Mathar, Feb 15 2010: (Start)

a(n) = 3*a(n-1) - 8*a(n-3) + 6*a(n-4) + 6*a(n-5) - 8*a(n-6) + 3*a(n-8) - a(n-9).

G.f.: x^2*(1+2*x+6*x^2+2*x^3+x^4) / ((1+x)^3*(x-1)^6). (End)

a(n) = Sum_{i=1..n} floor(i^2/4)^2). - Enrique Pérez Herrero, Mar 20 2012

a(n) = (2*n*(2*n^4+5*n^3-5*n+3) + 5*(2*n*(n+1)-1)*(-1)^n + 5)/320. - Bruno Berselli, Mar 21 2012

Maple

A059859 := n->add(A002620(i)^2, i=0..n);
f1 := n->1/160*(n-1)*(1+n)*(2*n^3+5*n^2+2*n-5); f2 := n->1/160*n*(n+2)*(2*n^3+n^2-2*n+4); A059859 := n-> if n mod 2 = 0 then f2(n) else f1(n); fi;

Mathematica

a[n_] := Sum[Floor[i^2/4]^2, {i, 1, n}]; Table[a[n], {n, 0, 100}] (* Enrique Pérez Herrero, Mar 20 2012 *)

Crossrefs

Cf. A002620.

Sequence in context: A299120 A033275 A166464 * A146617 A245240 A370839

Adjacent sequences: A059856 A059857 A059858 * A059860 A059861 A059862

Keyword

nonn,easy

Author

N. J. A. Sloane, Feb 26 2001

Status

approved