A212088 Number of (w,x,y,z) with all terms in {1,...,n} and w

0, 0, 7, 36, 117, 292, 612, 1143, 1963, 3159, 4833, 7099, 10080, 13914, 18751, 24750, 32085, 40942, 51516, 64017, 78667, 95697, 115353, 137893, 163584, 192708, 225559, 262440, 303669, 349576, 400500, 456795, 518827, 586971, 661617

Offset: 0

Author

Clark Kimberling, May 01 2012

Keywords

• nonn
• easy

Comments

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

a(n) + A212089(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, A212089.

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, 0, 7, 36, 117, 292, 612},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^2*(x^4+5*x^3+15*x^2+8*x+7) / ((x^2+x+1)*(1-x)^5). - Alois P. Heinz, May 18 2012