A211620 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and -1<=2w+x+y<=1.

0, 2, 16, 38, 76, 122, 184, 254, 340, 434, 544, 662, 796, 938, 1096, 1262, 1444, 1634, 1840, 2054, 2284, 2522, 2776, 3038, 3316, 3602, 3904, 4214, 4540, 4874, 5224, 5582, 5956, 6338, 6736, 7142, 7564, 7994, 8440, 8894, 9364, 9842, 10336, 10838, 11356, 11882

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,-2,1).

Crossrefs

Cf. A211422.

Programs

Mathematica

t = Compile[{{u, _Integer}},
   Module[{s = 0}, (Do[If[-1 <= 2 w + x + y <= 1,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 70]]  (* A211620 *)
%/2                        (* integers *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
Join[{0},LinearRecurrence[{2, 0, -2, 1},{2, 16, 38, 76},42]] (* Ray Chandler, Aug 02 2015 *)

PARI

concat(0, Vec(2*x*(1 + 6*x + 3*x^2 + 2*x^3) / ((1 - x)^3*(1 + x)) + O(x^40))) \\ Colin Barker, Dec 04 2017

Formula

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

From Colin Barker, Dec 04 2017: (Start)

G.f.: 2*x*(1 + 6*x + 3*x^2 + 2*x^3) / ((1 - x)^3*(1 + x)).

a(n) = 6*n^2 - 6*n + 4 for n>0 and even.

a(n) = 6*n^2 - 6*n + 2 for n odd.

(End)