A212249 Number of (w,x,y,z) with all terms in {1,...,n} and 3w

0, 1, 12, 63, 202, 496, 1034, 1923, 3289, 5280, 8062, 11820, 16761, 23110, 31111, 41030, 53151, 67777, 85233, 105862, 130026, 158109, 190513, 227659, 269990, 317967, 372070, 432801, 500680, 576246, 660060, 752701, 854767, 966878

Offset: 0

Author

Clark Kimberling, May 09 2012

Keywords

• nonn
• easy

Comments

a(n)+A212250(n) = 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.

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]]  (* A212249  *)
(* 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*(1+8*x+21*x^2+17*x^3+11*x^4+x^5)/((1+x+x^2)*(1-x)^5). - Bruno Berselli, Jun 05 2012

a(n) = (59*n^4 -10*n^3 +5*n^2 -6*n -8*((((n+1) mod 3) +(-1)^((n+1) mod 3))*(-1)^(n mod 3)))/72. - Bruno Berselli, Jun 05 2012