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A059859 Sum of squares of first n quarter-squares (A002620). 2
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 (list; graph; refs; listen; history; text; internal format)
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 A203233

Adjacent sequences:  A059856 A059857 A059858 * A059860 A059861 A059862

KEYWORD

nonn,easy

AUTHOR

N. J. A. Sloane, Feb 26 2001

STATUS

approved

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Last modified May 29 20:42 EDT 2020. Contains 334710 sequences. (Running on oeis4.)