A211623 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and -1<=w+2x+3y<=1.

0, 2, 12, 28, 54, 86, 128, 176, 234, 298, 372, 452, 542, 638, 744, 856, 978, 1106, 1244, 1388, 1542, 1702, 1872, 2048, 2234, 2426, 2628, 2836, 3054, 3278, 3512, 3752, 4002, 4258, 4524, 4796, 5078, 5366, 5664, 5968, 6282, 6602, 6932, 7268, 7614, 7966, 8328

Offset

0,2

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).

Formula

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

From Colin Barker, Dec 05 2017: (Start)

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

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

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

(End)

Mathematica

t = Compile[{{u, _Integer}},
   Module[{s = 0}, (Do[If[-1 <= w + 2 x + 3 y <= 1,
         s = s + 1], {w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 70]]  (* A211623 *)
%/2  (* integers *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
Join[{0}, LinearRecurrence[{2, 0, -2, 1}, {2, 12, 28, 54}, 43]] (* Ray Chandler, Aug 02 2015 *)

Prog

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

Crossrefs

Cf. A211422.

Sequence in context: A119201 A164876 A225291 * A034318 A338798 A345694

Adjacent sequences: A211620 A211621 A211622 * A211624 A211625 A211626

Keyword

nonn,easy

Author

Clark Kimberling, Apr 16 2012

Status

approved