A212252 Number of (w,x,y,z) with all terms in {1,...,n} and 3w=x+y+z+n+2.

0, 0, 0, 3, 11, 24, 45, 76, 117, 171, 240, 324, 426, 548, 690, 855, 1045, 1260, 1503, 1776, 2079, 2415, 2786, 3192, 3636, 4120, 4644, 5211, 5823, 6480, 7185, 7940, 8745, 9603, 10516, 11484, 12510, 13596, 14742, 15951, 17225, 18564, 19971

Offset: 0

Author

Clark Kimberling, May 15 2012

Keywords

• nonn
• easy

Comments

Also, the number of (w,x,y,z) with all terms in {1,...,n} and 3w=x+y+z-n-2.

For a guide to related sequences, see A211795.

Crossrefs

Cf. A211795, A212251.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[3 w == x + y + z + n + 2, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]  (* A212252 *)
(* Peter J. C. Moses, Apr 13 2012 *)
Table[2/9-n/2-n^2/3+5n^3/18-2/9Cos[2 n Pi/3] + 4Sin[2 n Pi/3]/9/Sqrt[3], {n, 0, 20}] (* Benedict W. J. Irwin, Sep 05 2016 *)

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).

From Benedict W. J. Irwin, Sep 05 2016: (Start)

a(n)=2/9-n/2-n^2/3+5*n^3/18-2/9*cos(2*n*Pi/3)+4*sin(2*n*Pi/3)/(9*sqrt(3)).

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

(End)