A211612 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+x+y>=0.

0, 4, 35, 117, 274, 530, 909, 1435, 2132, 3024, 4135, 5489, 7110, 9022, 11249, 13815, 16744, 20060, 23787, 27949, 32570, 37674, 43285, 49427, 56124, 63400, 71279, 79785, 88942, 98774, 109305, 120559, 132560, 145332, 158899, 173285, 188514, 204610, 221597

Offset: 0

Author

Clark Kimberling, Apr 16 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 (4,-6,4,-1).

Crossrefs

Cf. A211422.

Programs

Mathematica

t = Compile[{{u, _Integer}}, Module[{s = 0}, (Do[If[w + x + y >= 0, s = s + 1], {w, #}, {x, #}, {y, #}] &[Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 60]]  (* A211612 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{4, -6, 4, -1},{0, 4, 35, 117},36] (* Ray Chandler, Aug 02 2015 *)

PARI

concat(0, Vec(x*(4 + 19*x + x^2) / (1 - x)^4 + O(x^40))) \\ Colin Barker, Dec 04 2017

Formula

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

From Colin Barker, Dec 04 2017: (Start)

G.f.: x*(4 + 19*x + x^2) / (1 - x)^4.

a(n) = (n*(-3 + 3*n + 8*n^2))/2.

(End)