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A212521 Number of (w,x,y,z) with all terms in {1,...,n} and w>=2x and y<=3z.

%I A212521 #8 Dec 11 2015 08:08:22
%S A212521 0,0,4,18,60,138,297,528,912,1440,2200,3180,4536,6174,8330,10920,
%T A212521 14144,17928,22599,27900,34300,41580,50094,59664,70848,83148,97344,
%U A212521 113022,130732,150150,172125,195840,222464,251328,283220,317628,355752,396378,441142
%N A212521 Number of (w,x,y,z) with all terms in {1,...,n} and w>=2x and y<=3z.
%C A212521 For a guide to related sequences, see A211795.
%H A212521 Colin Barker, <a href="/A212521/b212521.txt">Table of n, a(n) for n = 0..1000</a>
%H A212521 <a href="/index/Rec#order_16">Index entries for linear recurrences with constant coefficients</a>, signature (0,2,2,-1,-4,0,2,0,-2,0,4,1,-2,-2,0,1).
%F A212521 a(n) = 2*a(n-2)+2*a(n-3)-a(n-4)-4*a(n-5)+2*a(n-7)-2*a(n-9)+4*a(n-11)+ a(n-12)-2*a(n-13)-2*a(n-14)+a(n-16).
%F A212521 G.f.: x^2*(4 +18*x +52*x^2 +94*x^3 +145*x^4 +166*x^5 +174*x^6 +160*x^7 +133*x^8 +80*x^9 +40*x^10 +12*x^11 +2*x^12) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3). - _Colin Barker_, Dec 11 2015
%t A212521 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212521 (Do[If[w >= 2 x && y <= 3 z, s = s + 1],
%t A212521 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212521 Map[t[#] &, Range[0, 50]]   (* A212521 *)
%o A212521 (PARI) concat([0,0], Vec(x^2*(4 +18*x +52*x^2 +94*x^3 +145*x^4 +166*x^5 +174*x^6 +160*x^7 +133*x^8 +80*x^9 +40*x^10 +12*x^11 +2*x^12) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3) + O(x^100))) \\ _Colin Barker_, Dec 11 2015
%Y A212521 Cf. A211795, A212508.
%K A212521 nonn,easy
%O A212521 0,3
%A A212521 _Clark Kimberling_, May 20 2012