A213483 Number of (w,x,y) with all terms in {0,...,n} and |w-x| + |x-y| >= w+x+y.

1, 5, 13, 23, 38, 55, 78, 103, 135, 169, 211, 255, 308, 363, 428, 495, 573, 653, 745, 839, 946, 1055, 1178, 1303, 1443, 1585, 1743, 1903, 2080, 2259, 2456, 2655, 2873, 3093, 3333, 3575, 3838, 4103, 4390, 4679, 4991, 5305, 5643, 5983, 6348

Offset: 0

Author

Clark Kimberling, Jun 13 2012

Keywords

• nonn
• easy

Comments

a(n) + A213482(n) = (n+1)^3.

For a guide to related sequences, see A212959.

Links

• Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,2,-1).

Crossrefs

Cf. A212959.

Programs

Mathematica

t = Compile[{{n, _Integer}}, Module[{s = 0},
(Do[If[w + x + y <= Abs[w - x] + Abs[x - y], s = s + 1],
{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
Map[t[#] &, Range[0, 60]]   (* A213483 *)
LinearRecurrence[{2,1,-4,1,2,-1},{1,5,13,23,38,55},50] (* Harvey P. Dale, Sep 11 2019 *)

Formula

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

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

From Ayoub Saber Rguez, Dec 31 2021: (Start)

a(n) + A213482(n) = (n+1)^3.

a(n) = A213479(n) + A006918(n).

a(n)= (n^3 + 33*n^2 + 71*n + 15 + (3*n+9)*((n+1) mod 2))/24. (End)