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

0, 0, 3, 11, 25, 48, 82, 128, 189, 267, 363, 480, 620, 784, 975, 1195, 1445, 1728, 2046, 2400, 2793, 3227, 3703, 4224, 4792, 5408, 6075, 6795, 7569, 8400, 9290, 10240, 11253, 12331, 13475, 14688, 15972, 17328, 18759, 20267, 21853, 23520

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, A212972.

Cf. A045991, 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]]   (* A212971*)
LinearRecurrence[{3,-3,2,-3,3,-1},{0,0,3,11,25,48},50] (* Harvey P. Dale, Aug 24 2021 *)

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

a(n) = (n+1)^3 - A212972(n).

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

a(n) = A045991(n+1) - A212974(n).

a(n) = (n^3 + n^2 - n - 1 + (((n+1) mod 3) mod 2))/3. (End)