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%I #9 Jan 29 2024 11:01:30
%S 1,2,6,2,8,17,2,8,18,36,2,8,18,32,65,2,8,18,32,50,106,2,8,18,32,50,72,
%T 161,2,8,18,32,50,72,98,232,2,8,18,32,50,72,98,128,321,2,8,18,32,50,
%U 72,98,128,162,430,2,8,18,32,50,72,98,128,162,200,561,2
%N Triangular array T, read by rows: T(n,k) = number of sums |x-y| + |y-z| - |x-z| = 2n-2-k, where x,y,z are in {1,2,...,n}.
%C The rows are the reversals of the rows in A368521.
%e First eight rows:
%e 1
%e 2 6
%e 2 8 17
%e 2 8 18 36
%e 2 8 18 32 65
%e 2 8 18 32 50 106
%e 2 8 18 32 50 72 161
%e 2 8 18 32 50 72 98 232
%e For n=2, there are 8 triples (x,y,z):
%e 111: |x-y| + |y-z| - |x-z| = 0
%e 112: |x-y| + |y-z| - |x-z| = 0
%e 121: |x-y| + |y-z| - |x-z| = 2
%e 122: |x-y| + |y-z| - |x-z| = 0
%e 211: |x-y| + |y-z| - |x-z| = 0
%e 212: |x-y| + |y-z| - |x-z| = 2
%e 221: |x-y| + |y-z| - |x-z| = 0
%e 222: |x-y| + |y-z| - |x-z| = 0
%e so row 2 of the array is (2,6), representing two 2s and six 0s.
%t t[n_] := t[n] = Tuples[Range[n], 3]
%t a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]]
%t - Abs[#[[1]] - #[[3]]] == 2n-2-k &]
%t u = Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2, 2}]
%t v = Flatten[u] (* sequence *)
%t Column[Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2, 2}]] (* array *)
%Y Cf. A084990 (column 1), A000578 (row sums), A001105 (limiting row), A368521.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Jan 25 2024
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