A212505 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y>=2z.

0, 0, 3, 14, 48, 114, 243, 444, 768, 1220, 1875, 2730, 3888, 5334, 7203, 9464, 12288, 15624, 19683, 24390, 30000, 36410, 43923, 52404, 62208, 73164, 85683, 99554, 115248, 132510, 151875, 173040, 196608, 222224, 250563, 281214, 314928

Offset: 0

Author

Clark Kimberling, May 19 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Crossrefs

Cf. A211795.

Programs

Maple

a:= n-> floor(n^2/4)*ceil(n^2*3/4):
seq(a(n), n=0..40);  # Alois P. Heinz, Aug 13 2013

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w < 2 x && y >= 2 z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212505 *)

Formula

a(n) = 2a(n-1)+2a(n-2)-6a(n-3)+6a(n-5)-2a(n-6)-2a(n-7)+a(n-8).

From Alois P. Heinz, May 31 2012: (Start)

a(n) = A077043(n) * A002620(n) = floor(n^2/4)*ceiling(n^2*3/4).

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