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%I #9 Dec 11 2015 11:01:50
%S 0,0,0,9,30,92,198,396,684,1152,1760,2650,3780,5292,7140,9555,12376,
%T 15936,20088,25110,30870,37800,45540,54692,64944,76752,89856,104949,
%U 121394,140140,160650,183600,208560,236544,266560,299982,335988,375516,417924,464607
%N Number of (w,x,y,z) with all terms in {1,...,n} and w>2x and y<=3z.
%C For a guide to related sequences, see A211795.
%H Colin Barker, <a href="/A212517/b212517.txt">Table of n, a(n) for n = 0..1000</a>
%H <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 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 G.f.: x^3*(9 +30*x +74*x^2 +120*x^3 +161*x^4 +170*x^5 +176*x^6 +148*x^7 +106*x^8 +58*x^9 +24*x^10 +4*x^11) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3). - _Colin Barker_, Dec 11 2015
%t t = Compile[{{n, _Integer}}, Module[{s = 0},
%t (Do[If[w > 2 x && y <= 3 z, s = s + 1],
%t {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t Map[t[#] &, Range[0, 50]] (* A212517 *)
%t FindLinearRecurrence[%]
%t LinearRecurrence[{0, 2, 2, -1, -4, 0, 2, 0, -2, 0, 4, 1, -2, -2, 0, 1},{0, 0, 0, 9, 30, 92, 198, 396, 684, 1152, 1760, 2650, 3780, 5292, 7140, 9555},37] (* _Ray Chandler_, Aug 02 2015 *)
%o (PARI) concat(vector(3), Vec(x^3*(9 +30*x +74*x^2 +120*x^3 +161*x^4 +170*x^5 +176*x^6 +148*x^7 +106*x^8 +58*x^9 +24*x^10 +4*x^11) / ((1 -x)^5*(1 +x)^3*(1 -x +x^2)*(1 +x +x^2)^3) + O(x^100))) \\ _Colin Barker_, Dec 11 2015
%Y Cf. A211795, A212508.
%K nonn,easy
%O 0,4
%A _Clark Kimberling_, May 20 2012
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