A368158 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.

2, 3, 1, 3, 6, 6, 2, 1, 4, 9, 11, 9, 4, 2, 1, 5, 12, 16, 16, 13, 6, 4, 2, 1, 6, 15, 21, 23, 22, 17, 9, 6, 4, 2, 1, 7, 18, 26, 30, 31, 28, 22, 12, 9, 6, 4, 2, 1, 8, 21, 31, 37, 40, 39, 35, 27, 16, 12, 9, 6, 4, 2, 1, 9, 24, 36, 44, 49, 50, 48, 42, 33, 20, 16

Offset: 1

Author

Clark Kimberling, Jan 20 2024

Keywords

• nonn
• tabf

Comments

Row n consists of 2n+1 positive integers.

Examples

			First six rows:
  2   3   1
  3   6   6   2   1
  4   9  11   9   4   2   1
  5  12  16  16  13   6   4   2  1
  6  15  21  23  22  17   9   6  4  2  1
  7  18  26  30  31  28  22  12  9  6  4  2  1
For n=2, there are 6 triples (x,y,z) having x <= y:
  111:  |x-y| + |y-z| = 0
  112:  |x-y| + |y-z| = 1
  121:  |x-y| + |y-z| = 2
  122:  |x-y| + |y-z| = 1
  221:  |x-y| + |y-z| = 1
  222:  |x-y| + |y-z| = 0,
so row 1 of the array is (2,3,1), representing two 0s, three 1s, and one 1.
		

Crossrefs

Cf. A002411 (row sums), A002620 (limiting reverse row), A368434, A368437, A368515, A368516, A368517, A368519, A368520, A368521, A368522.

Programs

Mathematica

t1[n_] := t1[n] = Tuples[Range[n], 3];
t[n_] := t[n] = Select[t1[n], #[[1]] < #[[2]] &];
a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
u = Table[Length[a[n, k]], {n, 2, 15}, {k, 0, 2 n - 2}];
v = Flatten[u]  (* sequence *)
Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 0, 2 n - 2}]]  (* array *)