A212963 a(n) = number of ordered triples (w,x,y) such that w,x,y are all in {0,...,n} and the numbers |w-x|, |x-y|, |y-w| are distinct.

0, 0, 0, 12, 36, 84, 156, 264, 408, 600, 840, 1140, 1500, 1932, 2436, 3024, 3696, 4464, 5328, 6300, 7380, 8580, 9900, 11352, 12936, 14664, 16536, 18564, 20748, 23100, 25620, 28320, 31200, 34272, 37536, 41004, 44676, 48564, 52668, 57000

Offset: 0

Author

Clark Kimberling, Jun 02 2012

Keywords

• nonn
• easy

Comments

For each n, there are (n+1)^3 ordered triples, ranging in lexicographical order from (0,0,0) to (n,n,n). For n = 3, the ordered triples (w,x,y) for which |w-x|, |x-y|, |y-w| are distinct are listed in the Example.

For a guide to related sequences, see A212959.

The ambiguous term "ordered triple" here means that the order matters: (w,x,y) is a different triple from (w,y,x), etc. It does not mean that wN. J. A. Sloane, Dec 28 2021

Examples

			a(3) counts the 12 ordered triples in the first column of the following list:
(w,x,y) (|w-x|,|x-y|,|y-w|)
----------------------------
(0,1,3)      (1,2,3)
(0,2,3)      (2,1,3)
(0,3,1)      (3,2,1)
(0,3,2)      (3,1,2)
(1,0,3)      (1,3,2)
(1,3,0)      (2,3,1)
(2,0,3)      (2,3,1)
(2,3,0)      (1,3,2)
(3,0,1)      (3,1,2)
(3,0,2)      (3,2,1)
(3,1,0)      (2,1,3)
(3,2,0)      (1,2,3)
		

Links

• Index entries for linear recurrences with constant coefficients, signature (3,-2,-2,3,-1).

Crossrefs

Cf. A002623, A212959.

Programs

Mathematica

t = Compile[{{n, _Integer}},
Module[{s = 0}, (Do[If[Abs[w - x] != Abs[x - y] && Abs[x - y] != Abs[y - w] &&
Abs[y - w] != Abs[w - x], s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
m = Map[t[#] &, Range[0, 45]]   (*A212963*)
m/12 (*essentially A002623*)

Formula

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

G.f.: 12*x^3/((1 + x)*(1 - x)^4).

a(n+3) = 12*A002623(n).

a(n) = (2*n^3 - 3*n^2 - 2*n + 3*(n mod 2))/2. - Ayoub Saber Rguez, Dec 06 2021

Extensions

Definition corrected by Clark Kimberling, Dec 28 2021