A212973 Number of triples (w,x,y) with all terms in {0,...,n} and w <= floor((x+y)/3).

1, 4, 12, 27, 50, 84, 131, 192, 270, 367, 484, 624, 789, 980, 1200, 1451, 1734, 2052, 2407, 2800, 3234, 3711, 4232, 4800, 5417, 6084, 6804, 7579, 8410, 9300, 10251, 11264, 12342, 13487, 14700, 15984, 17341, 18772, 20280, 21867, 23534

Offset: 0

Author

Clark Kimberling, Jun 03 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A212959.

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-3,2,-3,3,-1).

Crossrefs

Cf. A011379, A212959, A212972, A212974.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w <= Floor[(x + y)/3], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 60]]   (* A212973 *)
LinearRecurrence[{3,-3,2,-3,3,-1},{1,4,12,27,50,84},50] (* Harvey P. Dale, Jan 24 2015 *)

Formula

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).

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

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

From Ayoub Saber Rguez, Dec 11 2023: (Start)

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

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