A368604 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 and y > z.

0, 1, 0, 2, 2, 1, 3, 4, 4, 2, 4, 6, 7, 6, 4, 5, 8, 10, 10, 9, 6, 6, 10, 13, 14, 14, 12, 9, 7, 12, 16, 18, 19, 18, 16, 12, 8, 14, 19, 22, 24, 24, 23, 20, 16, 9, 16, 22, 26, 29, 30, 30, 28, 25, 20, 10, 18, 25, 30, 34, 36, 37, 36, 34, 30, 25, 11, 20, 28, 34, 39

Offset: 1

Author

Clark Kimberling, Jan 22 2024

Keywords

• nonn
• tabl

Comments

The difference sequence of the k-th column is (k), for k >= 1.

Examples

			First nine rows:
  0
  1    0
  2    2    1
  3    4    4    2
  4    6    7    6    4
  5    8   10   10    9    6
  6   10   13   14   14   12    9
  7   12   16   18   19   18   16   12
  8   14   19   22   24   24   23   20   16
For n=3, there are 5 triples (x,y,z) having x < y and y > z:
121:  |x-y| + |y-z| = 2
131:  |x-y| + |y-z| = 4
132:  |x-y| + |y-z| = 3
231:  |x-y| + |y-z| = 3
232:  |x-y| + |y-z| = 2
so that row 1 of the array is (2,2,1), representing two 2s, two 3s, and one 4.
		

Crossrefs

Cf. A000027 (column 1), A002620 (T(n,n)), A002717 (row sums), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368521, A368522.

Programs

Mathematica

t1[n_] := t1[n] = Tuples[Range[n], 3];
t[n_] := t[n] = Select[t1[n], #[[1]] < #[[2]] > #[[3]] &];
a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
u = Table[Length[a[n, k]], {n, 1, 15}, {k, 2, n + 1}];
v = Flatten[u]
Column[Table[Length[a[n, k]], {n, 1, 15}, {k, 2, n + 1}]]