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a(3) counts the 12 ordered triples in the first column of the following list:

(w,x,y) (|w-x|,|x-y|,|y-w|)

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(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)

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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.

|w-x|, |x-y|, |y-w| are distinct are listed in the Example.

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 w<x<y. - N. J. A. Sloane, Dec 28 2021

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

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.

t = Compile[{{n, _Integer}}, Module[{s = 0},

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)]];

{w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];

m = Map[t[#] &, Range[0, 45]] (* A212963 *)

m/12 (* essentially A002623 *)

Under construction, do not touch - N. J. A. Sloane, Dec 28 2021. I will ask Clark to clarify the definition. Are we assuming W < x < y? What precisely are the inequalites that are being assumed? Given that the definition mentioned "z", there is cause for suspicion. Please give examples for n = 3 and n = 4.

Definition corrected by Clark Kimberling, Dec 28 2021

Under construction, do not touch - N. J. A. Sloane, Dec 28 2021. I will ask Clark to clarify the definition. Are we assuming W < x < y? What precisely are the inequalites that are being assumed? Given that the definition mentioned "z", there is cause for suspicion. Please give examples for n = 3 and n = 4.

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Under construction, do not touch - N. J. A. Sloane, Dec 28 2021