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A212767 Number of (w,x,y,z) with all terms in {0,...,n}, w even, x even, and w+x=y+z.

%I A212767 #15 Jun 13 2015 00:54:14
%S A212767 1,1,8,10,29,35,72,84,145,165,256,286,413,455,624,680,897,969,1240,
%T A212767 1330,1661,1771,2168,2300,2769,2925,3472,3654,4285,4495,5216,5456,
%U A212767 6273,6545,7464,7770,8797,9139,10280,10660,11921,12341,13728,14190
%N A212767 Number of (w,x,y,z) with all terms in {0,...,n}, w even, x even, and w+x=y+z.
%C A212767 For a guide to related sequences, see A211795.
%H A212767 <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-3,-3,3,1,-1).
%F A212767 a(n) = a(n-1)+3*a(n-2)-3*a(n-3)-3*a(n-4)+3*a(n-5)+a(n-6)-a(n-7).
%F A212767 G.f.: ( 1+4*x^2+2*x^3+x^4 ) / ( (1+x)^3*(1-x)^4 ).
%F A212767 a(n) = (4*n^3+15*n^2+20*n+12+3*(n^2+4*n+4)*(-1)^n)/24. - _Luce ETIENNE_, Jun 03 2014
%t A212767 t = Compile[{{n, _Integer}}, Module[{s = 0}, (Do[If[(Mod[w, 2] == 0) && (Mod[x, 2] == 0 && w + x == y + z), s++], {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];
%t A212767 Map[t[#] &, Range[0, 50]]  (* A212767 *)
%t A212767 LinearRecurrence[{1, 3, -3, -3, 3, 1, -1}, {1, 1, 8, 10, 29, 35, 72}, 50]
%Y A212767 Cf. A211795.
%K A212767 nonn,easy
%O A212767 0,3
%A A212767 _Clark Kimberling_, May 29 2012