A211545 Number of ordered triples (w,x,y) with all terms in {-n,...-1,1,...,n} and w+x+y>0.

0, 4, 29, 99, 238, 470, 819, 1309, 1964, 2808, 3865, 5159, 6714, 8554, 10703, 13185, 16024, 19244, 22869, 26923, 31430, 36414, 41899, 47909, 54468, 61600, 69329, 77679, 86674, 96338, 106695, 117769, 129584, 142164, 155533, 169715, 184734, 200614, 217379

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 (4,-6,4,-1).

Formula

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

From Colin Barker, Dec 04 2017: (Start)

G.f.: x*(4 + 13*x + 7*x^2) / (1 - x)^4.

a(n) = (n*(3 - 3*n + 8*n^2))/2.

(End)

Example

a(1) counts these triples: (-1,1,1), (1,-1,1), (1,1,-1), (1,1,1).

Mathematica

t = Compile[{{u, _Integer}},
   Module[{s = 0}, (Do[If[w + x + y > 0, s = s + 1],
{w, #}, {x, #}, {y, #}] &[
      Flatten[{Reverse[-#], #} &[Range[1, u]]]]; s)]];
Map[t[#] &, Range[0, 60]]  (* A211545 *)
FindLinearRecurrence[%]
(* Peter J. C. Moses, Apr 13 2012 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 4, 29, 99}, 36] (* Ray Chandler, Aug 02 2015 *)

Prog

(PARI) concat(0, Vec(x*(4 + 13*x + 7*x^2) / (1 - x)^4 + O(x^40))) \\ Colin Barker, Dec 04 2017

Crossrefs

Cf. A211422.

Sequence in context: A288542 A024394 A199399 * A295842 A345897 A368372

Adjacent sequences: A211542 A211543 A211544 * A211546 A211547 A211548

Keyword

nonn,easy

Author

Clark Kimberling, Apr 16 2012

Status

approved