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A212963
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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.
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2
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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
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OFFSET
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0,4
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COMMENTS
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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.
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 w<x<y. - N. J. A. Sloane, Dec 28 2021
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LINKS
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FORMULA
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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).
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EXAMPLE
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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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MATHEMATICA
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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*)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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