A212506 Number of (w,x,y,z) with all terms in {1,...,n} and w<=2x and y<=2z.

0, 1, 16, 64, 196, 441, 900, 1600, 2704, 4225, 6400, 9216, 12996, 17689, 23716, 30976, 40000, 50625, 63504, 78400, 96100, 116281, 139876, 166464, 197136, 231361, 270400, 313600, 362404, 416025, 476100, 541696, 614656, 693889, 781456

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

a:= n-> (n*(n+1)/2+floor(n^2/4))^2:
seq(a(n), n=0..60);  # Alois P. Heinz, May 31 2012

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]]   (* A212506 *)

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) = A006578(n)^2.

G.f.: x*(14*x+30*x^2+42*x^3+17*x^4+4*x^5+1) / ((x+1)^3*(1-x)^5). (End)