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

1, 8, 24, 53, 100, 168, 261, 384, 540, 733, 968, 1248, 1577, 1960, 2400, 2901, 3468, 4104, 4813, 5600, 6468, 7421, 8464, 9600, 10833, 12168, 13608, 15157, 16820, 18600, 20501, 22528, 24684, 26973, 29400, 31968, 34681, 37544, 40560, 43733

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. A212959, A212971.

Cf. A011379, A212973.

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]]   (* A212972 *)

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 + 5x + 3*x^2 + 3*x^3)/((1 + x + x^2)*(1-x)^4).

a(n) = (n+1)^3 - A212971(n).

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

a(n) = A011379(n+1) - A212973(n).

a(n) = (2*n^3 + 8*n^2 + 10*n + 4 - (((n+1) mod 3) mod 2))/3. (End)

Extensions

Name corrected by Ayoub Saber Rguez, Jan 09 2024