A368517 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.

1, 1, 2, 4, 2, 1, 3, 7, 7, 4, 2, 1, 4, 10, 12, 11, 6, 4, 2, 1, 5, 13, 17, 18, 15, 9, 6, 4, 2, 1, 6, 16, 22, 25, 24, 20, 12, 9, 6, 4, 2, 1, 7, 19, 27, 32, 33, 31, 25, 16, 12, 9, 6, 4, 2, 1, 8, 22, 32, 39, 42, 42, 38, 31, 20, 16, 12, 9, 6, 4, 2, 1, 9, 25, 37

Offset: 1

Author

Clark Kimberling, Dec 31 2023

Keywords

• nonn
• tabf

Comments

Row n consists of 2n positive integers.

Examples

			First eight rows:
  1   1
  2   4   2   1
  3   7   7   4   2   1
  4  10  12  11   6   4   2   1
  5  13  17  18  15   9   6   4   2   1
  6  16  22  25  24  20  12   9   6   4   2  1
  7  19  27  32  33  31  25  16  12   9   6  4  2  1
  8  22  32  39  42  42  38  31  20  16  12  9  6  4  2  1
For n=3, there are 9 triples (x,y,z) having x < y:
  121:  |x-y| + |y-z| = 2
  122:  |x-y| + |y-z| = 1
  123:  |x-y| + |y-z| = 2
  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 that row 2 of the array is (2,4,2,1), representing two 1s, four 2s, two 3s, and one 4.
		

Crossrefs

Cf. A006002 (row sums), A002620 (limiting reverse row), A368434, A368437, A368515, A368516, A368518, A368519, A368520, A368521, A368522.

Programs

Mathematica

t1[n_] := t1[n] = Tuples[Range[n], 3];
t[n_] := t[n] = Select[t1[n], #[[1]] < #[[2]] &];
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 *)