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

Author

N. J. A. Sloane, Feb 26 2001

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A002620.

Programs

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 *)

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