A368516 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 and y != z.

2, 6, 4, 2, 10, 12, 8, 4, 2, 14, 20, 20, 12, 8, 4, 2, 18, 28, 32, 28, 18, 12, 8, 4, 2, 22, 36, 44, 44, 38, 24, 18, 12, 8, 4, 2, 26, 44, 56, 60, 58, 48, 32, 24, 18, 12, 8, 4, 2, 30, 52, 68, 76, 78, 72, 60, 40, 32, 24, 18, 12, 8, 4, 2, 34, 60, 80, 92, 98, 96

Offset: 1

Author

Clark Kimberling, Dec 31 2023

Keywords

• nonn
• tabf

Comments

Row n consists of 2n-1 even positive integers.

Examples

			First six rows:
   2
   6    4    2
  10   12    8    4    2
  14   20   20   12    8    4    2
  18   28   32   28   18   12    8    4   2
  22   36   44   44   38   24   18   12   8   4   2
For n=3, there are 12 triples (x,y,z) having x != y and y != z:
  121:  |x-y| + |y-z| = 2
  123:  |x-y| + |y-z| = 2
  131:  |x-y| + |y-z| = 4
  132:  |x-y| + |y-z| = 3
  212:  |x-y| + |y-z| = 2
  213:  |x-y| + |y-z| = 3
  231:  |x-y| + |y-z| = 3
  232:  |x-y| + |y-z| = 2
  312:  |x-y| + |y-z| = 3
  313:  |x-y| + |y-z| = 4
  321:  |x-y| + |y-z| = 2
  323:  |x-y| + |y-z| = 2,
so that row 2 of the array is (6,4,2), representing six 2s, four 3s, and two 4s.
		

Crossrefs

Cf. A011379 (row sums), A007590 (limiting reverse row), A368434, A368437, A368515, A368517, A368518, A368519, A368520, A368521, A368522.

Programs

Mathematica

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