A212089 Number of (w,x,y,z) with all terms in {1,...,n} and w>=average{x,y,z}.

0, 1, 9, 45, 139, 333, 684, 1258, 2133, 3402, 5167, 7542, 10656, 14647, 19665, 25875, 33451, 42579, 53460, 66304, 81333, 98784, 118903, 141948, 168192, 197917, 231417, 269001, 310987, 357705, 409500, 466726, 529749, 598950, 674719

Offset: 0

Author

Clark Kimberling, May 01 2012

Keywords

• nonn

Comments

Also, number of (w,x,y,z) with all terms in {1,...,n} and w<=average{x,y,z}.

a(n)+A212088(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, A212069, A212088.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[3 w >= x + y + z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 50]] (* A212088 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{4, -6, 5, -5, 6, -4, 1},{0, 1, 9, 45, 139, 333, 684},35] (* Ray Chandler, Aug 02 2015 *)

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+7*x^4+8*x^3+15*x^2+5*x) / ((x^2+x+1)*(-x+1)^5). - Alois P. Heinz, May 18 2012