A211613 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+x+y>1.

0, 1, 20, 78, 199, 407, 726, 1180, 1793, 2589, 3592, 4826, 6315, 8083, 10154, 12552, 15301, 18425, 21948, 25894, 30287, 35151, 40510, 46388, 52809, 59797, 67376, 75570, 84403, 93899, 104082, 114976, 126605, 138993, 152164, 166142, 180951, 196615, 213158

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 > 1, s = s + 1], {w, #}, {x, #}, {y, #}] &[Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 60]]  (* A211613 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
Join[{0},LinearRecurrence[{4, -6, 4, -1},{1, 20, 78, 199},35]] (* Ray Chandler, Aug 02 2015 *)

PARI

concat(0, Vec(x*(1 + 16*x + 4*x^2 + 3*x^3) / (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) for n > 4.

From Colin Barker, Dec 04 2017: (Start)

G.f.: x*(1 + 16*x + 4*x^2 + 3*x^3) / (1 - x)^4.

a(n) = (-6 + 9*n - 9*n^2 + 8*n^3)/2 for n > 0. (End)

E.g.f.: 3 + exp(x)*(4*x^3 + 15*x^2/2 + 4*x - 3). - Stefano Spezia, Jun 20 2025