A211624 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+2x+2y>0.

0, 4, 30, 104, 245, 485, 837, 1339, 1998, 2858, 3920, 5234, 6795, 8659, 10815, 13325, 16172, 19424, 23058, 27148, 31665, 36689, 42185, 48239, 54810, 61990, 69732, 78134, 87143, 96863, 107235, 118369, 130200, 142844, 156230, 170480, 185517, 201469, 218253

Offset: 0

Author

Clark Kimberling, Apr 17 2012

Keywords

• nonn
• easy

Comments

For a guide to related sequences, see A211422.

Links

• Colin Barker, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Cf. A211422.

Programs

Mathematica

t = Compile[{{u, _Integer}},
   Module[{s = 0}, (Do[If[w + 2 x + 2 y > 0,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 60]]  (* A211624 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{2, 1, -4, 1, 2, -1},{0, 4, 30, 104, 245, 485},36] (* Ray Chandler, Aug 02 2015 *)

PARI

concat(0, Vec(x*(4 + 22*x + 40*x^2 + 23*x^3 + 7*x^4) / ((1 - x)^4*(1 + x)^2) + O(x^50))) \\ Colin Barker, Aug 23 2017

Formula

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

From Colin Barker, Aug 23 2017: (Start)

G.f.: x*(4 + 22*x + 40*x^2 + 23*x^3 + 7*x^4) / ((1 - x)^4*(1 + x)^2).

a(n) = (64*n^3 - 14*n^2 + 12*n) / 16 for n even.

a(n) = (64*n^3 - 14*n^2 + 24*n - 10) / 16 for n odd. (End)