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%I #11 Dec 05 2015 09:40:49
%S 0,0,1,8,28,74,159,304,528,860,1325,1960,2796,3878,5243,6944,9024,
%T 11544,14553,18120,22300,27170,32791,39248,46608,54964,64389,74984,
%U 86828,100030,114675,130880,148736,168368,189873,213384,239004,266874,297103,329840
%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="/A212565/b212565.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*(14*n^4-12*n^3-8*n^2-6*((-1)^n-1)*n+3*((-1)^n-1)).
%F G.f.: x^2*(1+5*x+5*x^2+3*x^3) / ((1-x)^5*(1+x)^2).
%F (End)
%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]] (* A212565 *)
%o (PARI) concat([0,0], Vec(x^2*(1+5*x+5*x^2+3*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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