A212564 Number of (w,x,y,z) with all terms in {1,...,n} and w + x > 2y + 2z.

0, 0, 0, 3, 16, 48, 114, 229, 416, 696, 1100, 1655, 2400, 3368, 4606, 6153, 8064, 10384, 13176, 16491, 20400, 24960, 30250, 36333, 43296, 51208, 60164, 70239, 81536, 94136, 108150, 123665, 140800, 159648, 180336, 202963, 227664, 254544, 283746, 315381

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]]   (* A212564 *)
LinearRecurrence[{3,-1,-5,5,1,-3,1},{0,0,0,3,16,48,114},50] (* Harvey P. Dale, Apr 18 2023 *)

PARI

concat(vector(3), Vec(x^3*(3+7*x+3*x^2+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)*(2*n*(3*((-1)^n-1) + (n-2)*n*(7*n-4)) - 9*(-1)^n+9).

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

E.g.f.: (x*(7*x^3 + 24*x^2 + 3*x - 9)*cosh(x) + (7*x^4 + 24*x^3 + 3*x^2 - 3*x + 9)*sinh(x))/48. - Stefano Spezia, Jul 12 2023