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A212563 Number of (w,x,y,z) with all terms in {1,...,n} and w+x<=2y+2z.

%I A212563 #10 Aug 28 2020 11:10:29
%S A212563 0,1,16,78,240,577,1182,2172,3680,5865,8900,12986,18336,25193,33810,
%T A212563 44472,57472,73137,91800,113830,139600,169521,204006,243508,288480,
%U A212563 339417,396812,461202,533120,613145,701850,799856,907776,1026273,1156000,1297662,1451952
%N A212563 Number of (w,x,y,z) with all terms in {1,...,n} and w+x<=2y+2z.
%C A212563 For a guide to related sequences, see A211795.
%H A212563 Colin Barker, <a href="/A212563/b212563.txt">Table of n, a(n) for n = 0..1000</a>
%H A212563 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (3,-1,-5,5,1,-3,1).
%F A212563 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 A212563 From _Colin Barker_, Dec 10 2015: (Start)
%F A212563 a(n) = 1/96*(82*n^4+36*n^3-16*n^2-6*((-1)^n-1)*n+9*((-1)^n-1)).
%F A212563 G.f.: x*(1+13*x+31*x^2+27*x^3+10*x^4) / ((1-x)^5*(1+x)^2).
%F A212563 (End)
%t A212563 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212563 (Do[If[w + x <= 2 y + 2 z, s = s + 1],
%t A212563 {w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
%t A212563 Map[t[#] &, Range[0, 40]]   (* A212563 *)
%t A212563 LinearRecurrence[{3,-1,-5,5,1,-3,1},{0,1,16,78,240,577,1182},40] (* _Harvey P. Dale_, Aug 28 2020 *)
%o A212563 (PARI) concat(0, Vec(x*(1+13*x+31*x^2+27*x^3+10*x^4) / ((1-x)^5*(1+x)^2) + O(x^50))) \\ _Colin Barker_, Dec 10 2015
%Y A212563 Cf. A211795.
%K A212563 nonn,easy
%O A212563 0,3
%A A212563 _Clark Kimberling_, May 21 2012