A211621 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+2x+3y>0.

0, 3, 29, 103, 247, 484, 843, 1342, 2008, 2866, 3938, 5247, 6822, 8681, 10851, 13357, 16221, 19466, 23121, 27204, 31742, 36760, 42280, 48325, 54924, 62095, 69865, 78259, 87299, 97008, 107415, 118538, 130404, 143038, 156462, 170699, 185778, 201717, 218543

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 (2,0,-1,-1,0,2,-1).

Crossrefs

Cf. A211422.

Programs

Mathematica

t = Compile[{{u, _Integer}}, Module[{s = 0}, (Do[If[w + 2 x + 3 y > 0, s = s + 1], {w, #}, {x, #}, {y, #}] &[Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 70]]  (* A211621 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{2, 0, -1, -1, 0, 2, -1},{0, 3, 29, 103, 247, 484, 843},36] (* Ray Chandler, Aug 02 2015 *)

PARI

concat(0, Vec(x*(3 + 23*x + 45*x^2 + 44*x^3 + 22*x^4 + 7*x^5) / ((1 - x)^4*(1 + x)*(1 + x + x^2)) + O(x^40))) \\ Colin Barker, Dec 05 2017

Formula

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

G.f.: x*(3 + 23*x + 45*x^2 + 44*x^3 + 22*x^4 + 7*x^5) / ((1 - x)^4*(1 + x)*(1 + x + x^2)). - Colin Barker, Dec 05 2017