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A059859 Sum of squares of first n quarter-squares (A002620). 2

%I

%S 0,0,1,5,21,57,138,282,538,938,1563,2463,3759,5523,7924,11060,15156,

%T 20340,26901,35001,45001,57101,71742,89166,109902,134238,162799,

%U 195923,234339,278439,329064,386664,452200,526184,609705,703341

%N Sum of squares of first n quarter-squares (A002620).

%H Vincenzo Librandi, <a href="/A059859/b059859.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (3, 0, -8, 6, 6, -8, 0, 3, -1).

%F 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.

%F From _R. J. Mathar_, Feb 15 2010: (Start)

%F 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).

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

%F a(n) = Sum_{i=1..n} floor(i^2/4)^2). - _Enrique Pérez Herrero_, Mar 20 2012

%F 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

%p A059859 := n->add(A002620(i)^2,i=0..n);

%p 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;

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

%Y Cf. A002620.

%K nonn,easy

%O 0,4

%A _N. J. A. Sloane_, Feb 26 2001

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Last modified July 12 13:21 EDT 2020. Contains 335663 sequences. (Running on oeis4.)