A211622 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+2x+3y>1.

0, 3, 26, 94, 229, 457, 800, 1284, 1931, 2767, 3814, 5098, 6641, 8469, 10604, 13072, 15895, 19099, 22706, 26742, 31229, 36193, 41656, 47644, 54179, 61287, 68990, 77314, 86281, 95917, 106244, 117288, 129071, 141619, 154954, 169102, 184085, 199929, 216656

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 (3,-2,-2,3,-1).

Crossrefs

Cf. A211422.

Programs

Mathematica

t = Compile[{{u, _Integer}},
   Module[{s = 0}, (Do[If[w + 2 x + 3 y > 1,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 70]]  (* A211622 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
Join[{0},LinearRecurrence[{3, -2, -2, 3, -1},{3, 26, 94, 229, 457},35]] (* Ray Chandler, Aug 02 2015 *)

PARI

concat(0, Vec(x*(3 + 17*x + 22*x^2 + 5*x^3 + x^4) / ((1 - x)^4*(1 + x)) + O(x^40))) \\ Colin Barker, Dec 05 2017

Formula

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

From Colin Barker, Dec 05 2017: (Start)

G.f.: x*(3 + 17*x + 22*x^2 + 5*x^3 + x^4) / ((1 - x)^4*(1 + x)).

a(n) = (8*n^3 - 4*n^2 + 3*n - 2) / 2 for n>0 and even.

a(n) = (16*n^3 - 8*n^2 + 6*n - 2) / 4 for n odd.

(End)