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Number of (w,x,y,z) with all terms in {0,...,n} and 2w=floor((x+y+z)/2).
1

%I #13 Jan 09 2024 16:36:11

%S 1,4,13,32,63,108,171,256,365,500,665,864,1099,1372,1687,2048,2457,

%T 2916,3429,4000,4631,5324,6083,6912,7813,8788,9841,10976,12195,13500,

%U 14895,16384,17969,19652,21437,23328,25327,27436,29659,32000

%N Number of (w,x,y,z) with all terms in {0,...,n} and 2w=floor((x+y+z)/2).

%C For a guide to related sequences, see A211795.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (4,-7,8,-7,4,-1).

%F a(n)=4*a(n-1)-7*a(n-2)+8*a(n-3)-7*a(n-4)+4*a(n-5)-a(n-6).

%F G.f.: (1 + 4*x^2 + x^4)/((1 - x)^4 * (1 + x^2)).

%t t = Compile[{{n, _Integer}}, Module[{s = 0},

%t (Do[If[2 w == Floor[(x + y + z)/2], s = s + 1],

%t {w, 0, n}, {x, 0, n}, {y, 0, n}, {z, 0, n}]; s)]];

%t Map[t[#] &, Range[0, 45]] (* A212747 *)

%Y Cf. A211795.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, May 28 2012