A212090 Number of (w,x,y,z) with all terms in {1,...,n} and w

0, 1, 16, 80, 251, 610, 1261, 2331, 3970, 6351, 9670, 14146, 20021, 27560, 37051, 48805, 63156, 80461, 101100, 125476, 154015, 187166, 225401, 269215, 319126, 375675, 439426, 510966, 590905, 679876, 778535, 887561, 1007656, 1139545

Offset: 0

Author

Clark Kimberling, May 01 2012

Keywords

• nonn
• easy

Comments

a(n)+A000332(n+1) = n^4. For a guide to related sequences, see A211795.

Links

• Index entries for linear recurrences with constant coefficients, signature (5, -10, 10, -5, 1).

Crossrefs

Cf. A211795, A212091.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < x + y + z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212090 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{5, -10, 10, -5, 1},{0, 1, 16, 80, 251},34] (* Ray Chandler, Aug 02 2015 *)

Formula

a(n) = 5a(n-1)-10a(n-2)+10a(n-3)-5a(n-4)+a(n-5).

G.f.: -x*(1+11*x+10*x^2+x^3) / (x-1)^5.

a(n) = n*(-2+(1+(2+23*n)*n)*n)/24.