A212504 Number of (w,x,y,z) with all terms in {1,...,n} and w<2x and y>2z.

0, 0, 0, 7, 24, 76, 162, 333, 576, 976, 1500, 2275, 3240, 4572, 6174, 8281, 10752, 13888, 17496, 21951, 27000, 33100, 39930, 48037, 57024, 67536, 79092, 92443, 107016, 123676, 141750, 162225, 184320, 209152, 235824, 265591, 297432

Offset: 0

Author

Clark Kimberling, May 19 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

Links

• Index entries for linear recurrences with constant coefficients, signature (2,2,-6,0,6,-2,-2,1).

Crossrefs

Cf. A211795.

Programs

Maple

A212504:=n->ceil(3*n^2/4)*floor((n-1)^2/4); seq(A212504(n), n=0..40); # Wesley Ivan Hurt, Jan 24 2014

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]]   (* A212504 *)

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-1) = ceiling(n^2*3/4) * floor((n-1)^2/4).

G.f.: x^3*(7+10*x+14*x^2+4*x^3+x^4) / ((x+1)^3*(1-x)^5). (End)