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A212972 Number of triples (w,x,y) with all terms in {0,...,n} and w >= floor((x+y)/3).

%I A212972 #31 Jan 18 2024 09:59:03
%S A212972 1,8,24,53,100,168,261,384,540,733,968,1248,1577,1960,2400,2901,3468,
%T A212972 4104,4813,5600,6468,7421,8464,9600,10833,12168,13608,15157,16820,
%U A212972 18600,20501,22528,24684,26973,29400,31968,34681,37544,40560,43733
%N A212972 Number of triples (w,x,y) with all terms in {0,...,n} and w >= floor((x+y)/3).
%C A212972 For a guide to related sequences, see A212959.
%H A212972 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,3,-1).
%F A212972 a(n) = 3*a(n-1) - 3*a(n-2) + 2*a(n-3) - 3*a(n-4) + 3*a(n-5) - a(n-6).
%F A212972 G.f.: (1 + 5x + 3*x^2 + 3*x^3)/((1 + x + x^2)*(1-x)^4).
%F A212972 a(n) = (n+1)^3 - A212971(n).
%F A212972 From _Ayoub Saber Rguez_, Dec 11 2023: (Start)
%F A212972 a(n) = A011379(n+1) - A212973(n).
%F A212972 a(n) = (2*n^3 + 8*n^2 + 10*n + 4 - (((n+1) mod 3) mod 2))/3. (End)
%t A212972 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212972 (Do[If[w >= Floor[(x + y)/3], s = s + 1],
%t A212972 {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
%t A212972 m = Map[t[#] &, Range[0, 60]]   (* A212972 *)
%Y A212972 Cf. A212959, A212971.
%Y A212972 Cf. A011379, A212973.
%K A212972 nonn,easy
%O A212972 0,2
%A A212972 _Clark Kimberling_, Jun 03 2012
%E A212972 Name corrected by _Ayoub Saber Rguez_, Jan 09 2024