A368605 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.

1, 1, 2, 3, 2, 1, 3, 5, 5, 4, 2, 1, 4, 7, 8, 8, 6, 4, 2, 1, 5, 9, 11, 12, 11, 9, 6, 4, 2, 1, 6, 11, 14, 16, 16, 15, 12, 9, 6, 4, 2, 1, 7, 13, 17, 20, 21, 21, 19, 16, 12, 9, 6, 4, 2, 1, 8, 15, 20, 24, 26, 27, 26, 24, 20, 16, 12, 9, 6, 4, 2, 1, 9, 17, 23, 28

Offset: 1

Author

Clark Kimberling, Jan 22 2024

Keywords

• nonn
• tabf

Comments

Row n consists of 2n positive integers.

Examples

			First six rows:
  1    1
  2    3    2    1
  3    5    5    4    2    1
  4    7    8    8    6    4    2   1
  5    9   11   12   11    9    6   4   2   1
  6   11   14   16   16   15   12   9   6   4   2   1
For n=3, there are 8 triples (x,y,z) having x < y and y >= z:
  121:  |x-y| + |y-z| = 2
  122:  |x-y| + |y-z| = 1
  131:  |x-y| + |y-z| = 4
  132:  |x-y| + |y-z| = 3
  133:  |x-y| + |y-z| = 2
  231:  |x-y| + |y-z| = 3
  232:  |x-y| + |y-z| = 2
  233:  |x-y| + |y-z| = 1
so row 1 of the array is (2,3,2,1), representing two 1s, three 2s, two 3s, and one 4.
		

Crossrefs

Cf. A000027 (column 1), A007290 (row sums), A002620 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368521, A368522, A368604, A368606, A368607, A368609.

Programs

Mathematica

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