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Number of triples (w,x,y) with all terms in {0,...,n} and w <= floor((x+y)/3).
4

%I #29 Jan 18 2024 09:59:16

%S 1,4,12,27,50,84,131,192,270,367,484,624,789,980,1200,1451,1734,2052,

%T 2407,2800,3234,3711,4232,4800,5417,6084,6804,7579,8410,9300,10251,

%U 11264,12342,13487,14700,15984,17341,18772,20280,21867,23534

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

%C For a guide to related sequences, see A212959.

%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (3,-3,2,-3,3,-1).

%F 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 G.f.: (1 + x + 3*x^2 + x^3)/((1+x+x^2)*(1-x)^4).

%F a(n) = (n+1)^3 - A212974(n).

%F From _Ayoub Saber Rguez_, Dec 11 2023: (Start)

%F a(n) = A011379(n+1) - A212972(n).

%F a(n) = (n^3 + 4*n^2 + 5*n + 2 + (((n+1) mod 3) mod 2))/3. (End)

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

%t (Do[If[w <= Floor[(x + y)/3], s = s + 1],

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

%t m = Map[t[#] &, Range[0, 60]] (* A212973 *)

%t LinearRecurrence[{3,-3,2,-3,3,-1},{1,4,12,27,50,84},50] (* _Harvey P. Dale_, Jan 24 2015 *)

%Y Cf. A011379, A212959, A212972, A212974.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_, Jun 03 2012