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%I #6 Jan 20 2024 09:46:25
%S 1,1,2,4,2,1,3,7,7,4,2,1,4,10,12,11,6,4,2,1,5,13,17,18,15,9,6,4,2,1,6,
%T 16,22,25,24,20,12,9,6,4,2,1,7,19,27,32,33,31,25,16,12,9,6,4,2,1,8,22,
%U 32,39,42,42,38,31,20,16,12,9,6,4,2,1,9,25,37
%N Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y|+|y-z| = k, where x,y,z are in {1,2,...,n} and x < y.
%C Row n consists of 2n positive integers.
%e First eight rows:
%e 1 1
%e 2 4 2 1
%e 3 7 7 4 2 1
%e 4 10 12 11 6 4 2 1
%e 5 13 17 18 15 9 6 4 2 1
%e 6 16 22 25 24 20 12 9 6 4 2 1
%e 7 19 27 32 33 31 25 16 12 9 6 4 2 1
%e 8 22 32 39 42 42 38 31 20 16 12 9 6 4 2 1
%e For n=3, there are 9 triples (x,y,z) having x < y:
%e 121: |x-y| + |y-z| = 2
%e 122: |x-y| + |y-z| = 1
%e 123: |x-y| + |y-z| = 2
%e 131: |x-y| + |y-z| = 4
%e 132: |x-y| + |y-z| = 3
%e 133: |x-y| + |y-z| = 2
%e 231: |x-y| + |y-z| = 3
%e 232: |x-y| + |y-z| = 2
%e 233: |x-y| + |y-z| = 1,
%e so that row 2 of the array is (2,4,2,1), representing two 1s, four 2s, two 3s, and one 4.
%t t1[n_] := t1[n] = Tuples[Range[n], 3];
%t t[n_] := t[n] = Select[t1[n], #[[1]] < #[[2]] &];
%t a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
%t u = Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}];
%t v = Flatten[u] (* sequence *)
%t Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 2}]] (* array *)
%Y Cf. A006002 (row sums), A002620 (limiting reverse row), A368434, A368437, A368515, A368516, A368518, A368519, A368520, A368521, A368522.
%K nonn,tabf
%O 1,3
%A _Clark Kimberling_, Dec 31 2023
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