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%I #17 Jul 14 2023 12:10:33
%S 0,0,0,3,16,48,114,229,416,696,1100,1655,2400,3368,4606,6153,8064,
%T 10384,13176,16491,20400,24960,30250,36333,43296,51208,60164,70239,
%U 81536,94136,108150,123665,140800,159648,180336,202963,227664,254544,283746,315381
%N Number of (w,x,y,z) with all terms in {1,...,n} and w + x > 2y + 2z.
%C For a guide to related sequences, see A211795.
%H Colin Barker, <a href="/A212564/b212564.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).
%F a(n) = 3*a(n-1) - a(n-2) - 5*a(n-3) + 5*a(n-4) + a(n-5) - 3*a(n-6) + a(n-7).
%F From _Colin Barker_, Dec 05 2015: (Start)
%F a(n) = (1/96)*(2*n*(3*((-1)^n-1) + (n-2)*n*(7*n-4)) - 9*(-1)^n+9).
%F G.f.: x^3*(3+7*x+3*x^2+x^3) / ((1-x)^5*(1+x)^2). (End)
%F E.g.f.: (x*(7*x^3 + 24*x^2 + 3*x - 9)*cosh(x) + (7*x^4 + 24*x^3 + 3*x^2 - 3*x + 9)*sinh(x))/48. - _Stefano Spezia_, Jul 12 2023
%t t = Compile[{{n, _Integer}}, Module[{s = 0},
%t (Do[If[w + x > 2 y + 2 z, s = s + 1],
%t {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t Map[t[#] &, Range[0, 40]] (* A212564 *)
%t LinearRecurrence[{3,-1,-5,5,1,-3,1},{0,0,0,3,16,48,114},50] (* _Harvey P. Dale_, Apr 18 2023 *)
%o (PARI) concat(vector(3), Vec(x^3*(3+7*x+3*x^2+x^3) / ((1-x)^5*(1+x)^2) + O(x^100))) \\ _Colin Barker_, Dec 05 2015
%Y Cf. A211795.
%K nonn,easy
%O 0,4
%A _Clark Kimberling_, May 21 2012
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