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A368522 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}.

%I A368522 #9 Jan 29 2024 11:01:30
%S A368522 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 A368522 161,2,8,18,32,50,72,98,232,2,8,18,32,50,72,98,128,321,2,8,18,32,50,
%U A368522 72,98,128,162,430,2,8,18,32,50,72,98,128,162,200,561,2
%N A368522 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 A368522 The rows are the reversals of the rows in A368521.
%e A368522 First eight rows:
%e A368522    1
%e A368522    2   6
%e A368522    2   8   17
%e A368522    2   8   18   36
%e A368522    2   8   18   32   65
%e A368522    2   8   18   32   50   106
%e A368522    2   8   18   32   50    72   161
%e A368522    2   8   18   32   50    72    98    232
%e A368522 For n=2, there are 8 triples (x,y,z):
%e A368522   111:  |x-y| + |y-z| - |x-z| = 0
%e A368522   112:  |x-y| + |y-z| - |x-z| = 0
%e A368522   121:  |x-y| + |y-z| - |x-z| = 2
%e A368522   122:  |x-y| + |y-z| - |x-z| = 0
%e A368522   211:  |x-y| + |y-z| - |x-z| = 0
%e A368522   212:  |x-y| + |y-z| - |x-z| = 2
%e A368522   221:  |x-y| + |y-z| - |x-z| = 0
%e A368522   222:  |x-y| + |y-z| - |x-z| = 0
%e A368522 so row 2 of the array is (2,6), representing two 2s and six 0s.
%t A368522 t[n_] := t[n] = Tuples[Range[n], 3]
%t A368522 a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]]
%t A368522  - Abs[#[[1]] - #[[3]]] ==  2n-2-k &]
%t A368522 u = Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2, 2}]
%t A368522 v = Flatten[u] (* sequence *)
%t A368522 Column[Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2, 2}]]  (* array *)
%Y A368522 Cf. A084990 (column 1), A000578 (row sums), A001105 (limiting row), A368521.
%K A368522 nonn,tabl
%O A368522 1,2
%A A368522 _Clark Kimberling_, Jan 25 2024