A368436 Irregular triangular array T, read by rows: T(n,k) = number of sums |x-y|+|y-z| = k, where (x,y,z) is a permutation of three distinct numbers x,y,z taken from {0,1,...,n}, for n >= 2, k >= 2.

2, 4, 4, 12, 4, 4, 6, 20, 14, 12, 4, 4, 8, 28, 24, 28, 12, 12, 4, 4, 10, 36, 34, 44, 30, 24, 12, 12, 4, 4, 12, 44, 44, 60, 48, 48, 24, 24, 12, 12, 4, 4, 14, 52, 54, 76, 66, 72, 50, 40, 24, 24, 12, 12, 4, 4, 16, 60, 64, 92, 84, 96, 76, 72, 40, 40, 24, 24, 12

Offset: 1

Author

Clark Kimberling, Dec 25 2023

Keywords

• nonn
• tabf

Comments

Row n consists of 2n even positive integers having sum A007531(n+2) = (n+2)!/(n-1)!.

Examples

			Taking n = 2, the permutations of {x,y,z} of {0,1,2} with sums |x-y| + |y-z| = k, for k = 2,3, are as follows:
012: |0-1| + |1-2| = 2
021: |0-2| + |2-1| = 3
102: |1-0| + |0-2| = 3
120: |1-2| + |2-0| = 3
201: |2-0| + |0-1| = 3
210: |2-1| + |1-0| = 2
so that row 1 of the array is (2,4), representing two 2s and four 3s.
First eight rows:
 2    4
 4   12    4    4
 6   20   14   12    4    4
 8   28   24   28   12   12    4    4
10   36   34   44   30   24   12   12    4    4
12   44   44   60   48   48   24   24   12   12    4    4
14   52   54   76   66   72   50   40   24   24   12   12    4    4
16   60   64   92   84   96   76   72   40   40   24   24   12   12   4   4
		

Crossrefs

Cf. A007531, A368434, A368437 (reversed rows).

Programs

Mathematica

t[n_] := t[n] = Permutations[-1 + Range[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, 2, 2 n - 1}];
v = Flatten[u]  (* sequence *)
Column[u]       (* array *)