A368519 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.

2, 4, 3, 2, 6, 6, 8, 2, 2, 8, 9, 14, 9, 6, 2, 2, 10, 12, 20, 16, 16, 6, 6, 2, 2, 12, 15, 26, 23, 26, 17, 12, 6, 6, 2, 2, 14, 18, 32, 30, 36, 28, 26, 12, 12, 6, 6, 2, 2, 16, 21, 38, 37, 46, 39, 40, 27, 20, 12, 12, 6, 6, 2, 2, 18, 24, 44, 44, 56, 50, 54, 42

Offset: 1

Author

Clark Kimberling, Jan 22 2024

Keywords

• nonn
• tabf

Comments

Row n consists of 2n-1 positive integers.

Examples

			First six rows:
   2
   4   3    2
   6   6    8   2   2
   8   9   14   9   6   2   2
   10  12  20  16  16   6   6  2  2
   12  15  26  23  26  17  12  6  6  2  2
For n=3, there are 9 triples (x,y,z) having x < z:
112:  |x-y| + |y-z| = 1
113:  |x-y| + |y-z| = 2
122:  |x-y| + |y-z| = 1
123:  |x-y| + |y-z| = 2
132:  |x-y| + |y-z| = 3
133:  |x-y| + |y-z| = 2
213:  |x-y| + |y-z| = 3
223:  |x-y| + |y-z| = 1
233:  |x-y| + |y-z| = 1,
so that row 1 of the array is (4,3,2), representing four 1s, three 2s, and two 3s.
		

Crossrefs

Cf. A006002 (row sums), A110660 (limiting reverse row), A368434, A368437, A368515, A368516, A368517, A368518, A368520, 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, 2, 15}, {k, 1, 2 n - 3}];
v = Flatten[u];  (* sequence *)
Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 1, 2 n - 3}]]  (* array *)