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A212971 Number of triples (w,x,y) with all terms in {0,...,n} and w < floor((x+y)/3).

%I A212971 #32 Jan 18 2024 09:59:10
%S A212971 0,0,3,11,25,48,82,128,189,267,363,480,620,784,975,1195,1445,1728,
%T A212971 2046,2400,2793,3227,3703,4224,4792,5408,6075,6795,7569,8400,9290,
%U A212971 10240,11253,12331,13475,14688,15972,17328,18759,20267,21853,23520
%N A212971 Number of triples (w,x,y) with all terms in {0,...,n} and w < floor((x+y)/3).
%C A212971 For a guide to related sequences, see A212959.
%H A212971 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,3,-1).
%F A212971 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 A212971 G.f.: (x^2)*(3 + 2*x + x^2)/((1 + x + x^2)*(1-x)^4).
%F A212971 a(n) = (n+1)^3 - A212972(n).
%F A212971 From _Ayoub Saber Rguez_, Dec 11 2023: (Start)
%F A212971 a(n) = A045991(n+1) - A212974(n).
%F A212971 a(n) = (n^3 + n^2 - n - 1 + (((n+1) mod 3) mod 2))/3. (End)
%t A212971 t = Compile[{{n, _Integer}}, Module[{s = 0},
%t A212971 (Do[If[w < Floor[(x + y)/3], s = s + 1],
%t A212971 {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
%t A212971 m = Map[t[#] &, Range[0, 60]]   (* A212971*)
%t A212971 LinearRecurrence[{3,-3,2,-3,3,-1},{0,0,3,11,25,48},50] (* _Harvey P. Dale_, Aug 24 2021 *)
%Y A212971 Cf. A212959, A212972.
%Y A212971 Cf. A045991, A212974.
%K A212971 nonn,easy
%O A212971 0,3
%A A212971 _Clark Kimberling_, Jun 03 2012