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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.

%I A212963 #31 Dec 29 2021 10:33:36
%S A212963 0,0,0,12,36,84,156,264,408,600,840,1140,1500,1932,2436,3024,3696,
%T A212963 4464,5328,6300,7380,8580,9900,11352,12936,14664,16536,18564,20748,
%U A212963 23100,25620,28320,31200,34272,37536,41004,44676,48564,52668,57000
%N 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.
%C A212963 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.
%C A212963 For a guide to related sequences, see A212959.
%C A212963 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
%H A212963 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F A212963 a(n) = 3*a(n-1) - 2*a(n-2) - 2*a(n-3) + 3*a(n-4) - a(n-5).
%F A212963 G.f.: 12*x^3/((1 + x)*(1 - x)^4).
%F A212963 a(n+3) = 12*A002623(n).
%F A212963 a(n) = (2*n^3 - 3*n^2 - 2*n + 3*(n mod 2))/2. - _Ayoub Saber Rguez_, Dec 06 2021
%e A212963 a(3) counts the 12 ordered triples in the first column of the following list:
%e A212963 (w,x,y) (|w-x|,|x-y|,|y-w|)
%e A212963 ----------------------------
%e A212963 (0,1,3)      (1,2,3)
%e A212963 (0,2,3)      (2,1,3)
%e A212963 (0,3,1)      (3,2,1)
%e A212963 (0,3,2)      (3,1,2)
%e A212963 (1,0,3)      (1,3,2)
%e A212963 (1,3,0)      (2,3,1)
%e A212963 (2,0,3)      (2,3,1)
%e A212963 (2,3,0)      (1,3,2)
%e A212963 (3,0,1)      (3,1,2)
%e A212963 (3,0,2)      (3,2,1)
%e A212963 (3,1,0)      (2,1,3)
%e A212963 (3,2,0)      (1,2,3)
%t A212963 t = Compile[{{n, _Integer}},
%t A212963 Module[{s = 0}, (Do[If[Abs[w - x] != Abs[x - y] && Abs[x - y] != Abs[y - w] &&
%t A212963 Abs[y - w] != Abs[w - x], s = s + 1], {w, 0, n}, {x, 0, n}, {y, 0, n}]; s)]];
%t A212963 m = Map[t[#] &, Range[0, 45]]   (*A212963*)
%t A212963 m/12 (*essentially A002623*)
%Y A212963 Cf. A002623, A212959.
%K A212963 nonn,easy
%O A212963 0,4
%A A212963 _Clark Kimberling_, Jun 02 2012
%E A212963 Definition corrected by _Clark Kimberling_, Dec 28 2021