A211629 Number of ordered triples (w,x,y) with all terms in {-n, ..., -1, 1, ..., n} and 4w + x + y > 0.

0, 4, 31, 105, 252, 492, 851, 1353, 2024, 2884, 3959, 5273, 6852, 8716, 10891, 13401, 16272, 19524, 23183, 27273, 31820, 36844, 42371, 48425, 55032, 62212, 69991, 78393, 87444, 97164, 107579, 118713, 130592, 143236, 156671, 170921, 186012, 201964, 218803

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 (3,-3,1,1,-3,3,-1).

Crossrefs

Cf. A211422.

Programs

Mathematica

t = Compile[{{u, _Integer}},
   Module[{s = 0}, (Do[If[4 w + x + y > 0,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 60]]  (* A211629 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{3, -3, 1, 1, -3, 3, -1},{0, 4, 31, 105, 252, 492, 851},36] (* Ray Chandler, Aug 02 2015 *)

PARI

concat(0, Vec(x*(4 + 19*x + 24*x^2 + 26*x^3 + 16*x^4 + 7*x^5) / ((1 - x)^4*(1 + x)*(1 + x^2)) + O(x^40))) \\ Colin Barker, Dec 05 2017

Formula

a(n) = 3*a(n-1) - 3*a(n-2) + a(n-3) + a(n-4) - 3*a(n-5) + 3*a(n-6) - a(n-7) for n > 6.

G.f.: x*(4 + 19*x + 24*x^2 + 26*x^3 + 16*x^4 + 7*x^5) / ((1 - x)^4*(1 + x)*(1 + x^2)). - Colin Barker, Dec 05 2017