A368521 Triangular array T, read by rows: T(n,k) = number of sums |x-y| + |y-z| - |x-z| = k, where x,y,z are in {1,2,...,n}.

1, 6, 2, 17, 8, 2, 36, 18, 8, 2, 65, 32, 18, 8, 2, 106, 50, 32, 18, 8, 2, 161, 72, 50, 32, 18, 8, 2, 232, 98, 72, 50, 32, 18, 8, 2, 321, 128, 98, 72, 50, 32, 18, 8, 2, 430, 162, 128, 98, 72, 50, 32, 18, 8, 2, 561, 200, 162, 128, 98, 72, 50, 32, 18, 8, 2, 716

Offset: 1

Author

Clark Kimberling, Jan 25 2024

Keywords

• nonn
• tabl

Examples

			First eight rows:
   1
   6    2
  17    8    2
  36   18    8    2
  65   32   18    8    2
 106   50   32   18    8    2
 161   72   50   32   18    8    2
 232   98   72   50   32   18    8    2
For n=2, there are 8 triples (x,y,z):
  111:  |x-y| + |y-z| - |x-z| = 0
  112:  |x-y| + |y-z| - |x-z| = 0
  121:  |x-y| + |y-z| - |x-z| = 2
  122:  |x-y| + |y-z| - |x-z| = 0
  211:  |x-y| + |y-z| - |x-z| = 0
  212:  |x-y| + |y-z| - |x-z| = 2
  221:  |x-y| + |y-z| - |x-z| = 0
  222:  |x-y| + |y-z| - |x-z| = 0
so row 2 of the array is (6,2), representing six 0s and two 2s.
		

Crossrefs

Cf. A084990 (column 1), A000578 (row sums), A001105 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368522, A368604, A368605, A368606, A368607, A368609.

Programs

Mathematica

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