A212250 Number of (w,x,y,z) with all terms in {1,...,n} and 3w>=x+y+z+n.

0, 0, 4, 18, 54, 129, 262, 478, 807, 1281, 1938, 2821, 3975, 5451, 7305, 9595, 12385, 15744, 19743, 24459, 29974, 36372, 43743, 52182, 61786, 72658, 84906, 98640, 113976, 131035, 149940, 170820, 193809, 219043, 246664, 276819, 309657

Offset: 0

Author

Clark Kimberling, May 09 2012

Keywords

• nonn
• easy

Comments

a(n)+A212249 = n^4.

For a guide to related sequences, see A211795.

Links

• Index entries for linear recurrences with constant coefficients, signature (4,-6,5,-5,6,-4,1).

Crossrefs

Cf. A211795, A212247, A212249.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[3 w >= x + y + z + n, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]  (* A212250 *)
(* Peter J. C. Moses, Apr 13 2012 *)

Formula

a(n) = 4*a(n-1)-6*a(n-2)+5*a(n-3)-5*a(n-4)+6*a(n-5)-4*a(n-6)+a(n-7).

G.f.: x^2*(4+2*x+6*x^2+x^3)/((1+x+x^2)*(1-x)^5). - Bruno Berselli, Jun 05 2012

a(n) = (13*n^4+10*n^3-5*n^2+6*n+8*b)/72, where b = 0,-3,1,0,-3,1,... (repeated). [Bruno Berselli, Jun 05 2012]