A368520 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 <= z.

1, 2, 2, 2, 3, 4, 7, 2, 2, 4, 6, 12, 8, 6, 2, 2, 5, 8, 17, 14, 15, 6, 6, 2, 2, 6, 10, 22, 20, 24, 16, 12, 6, 6, 2, 2, 7, 12, 27, 26, 33, 26, 25, 12, 12, 6, 6, 2, 2, 8, 14, 32, 32, 42, 36, 38, 26, 20, 12, 12, 6, 6, 2, 2, 9, 16, 37, 38, 51, 46, 51, 40, 37, 20, 20

Offset: 1

Author

Clark Kimberling, Jan 22 2024

Keywords

• nonn
• tabf

Comments

Row n consists of 2n-1 positive integers.

Examples

			First seven rows:
 1
 2   2   2
 3   4   7   2   2
 4   6  12   8   6   2   2
 5   8  17  14  15   6   6   2   2
 6  10  22  20  24  16  12   6   6  2  2
 7  12  27  26  33  26  25  12  12  6  6  2  2
For n=2, there are 6 triples (x,y,z) having x <= z:
111:  |x-y| + |y-z| = 0
112:  |x-y| + |y-z| = 1
121:  |x-y| + |y-z| = 2
122:  |x-y| + |y-z| = 1
212:  |x-y| + |y-z| = 2
222:  |x-y| + |y-z| = 0
so that row 1 of the array is (2,2,2), representing two 0s, two 1s, and two 2s.
		

Crossrefs

Cf. A002411 (row sums), A110660 (limiting reverse row), 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]] <= #[[3]] &];
a[n_, k_] := Select[t[n], Abs[#[[1]] - #[[2]]] + Abs[#[[2]] - #[[3]]] == k &];
u = Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2}];
v = Flatten[u]  (* sequence *)
Column[Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2}]]  (* array *)