A212565 Number of (w,x,y,z) with all terms in {1,...,n} and w+x>=2y+2z.

0, 0, 1, 8, 28, 74, 159, 304, 528, 860, 1325, 1960, 2796, 3878, 5243, 6944, 9024, 11544, 14553, 18120, 22300, 27170, 32791, 39248, 46608, 54964, 64389, 74984, 86828, 100030, 114675, 130880, 148736, 168368, 189873, 213384, 239004, 266874, 297103, 329840

Offset: 0

Author

Clark Kimberling, May 21 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211795.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-5,5,1,-3,1).

Crossrefs

Cf. A211795.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x >= 2 y + 2 z, s = s + 1],
{w, 1, #}, {x, 1, #}, {y, 1, #}, {z, 1, #}] &[n]; s)]];
Map[t[#] &, Range[0, 40]]   (* A212565 *)

PARI

concat([0,0], Vec(x^2*(1+5*x+5*x^2+3*x^3)/((1-x)^5*(1+x)^2) + O(x^100))) \\ Colin Barker, Dec 05 2015

Formula

a(n) = 3*a(n-1)-a(n-2)-5*a(n-3)+5*a(n-4)+a(n-5)-3*a(n-6)+a(n-7).

From Colin Barker, Dec 05 2015: (Start)

a(n) = 1/96*(14*n^4-12*n^3-8*n^2-6*((-1)^n-1)*n+3*((-1)^n-1)).

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

(End)