A368158 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.

2, 3, 1, 3, 6, 6, 2, 1, 4, 9, 11, 9, 4, 2, 1, 5, 12, 16, 16, 13, 6, 4, 2, 1, 6, 15, 21, 23, 22, 17, 9, 6, 4, 2, 1, 7, 18, 26, 30, 31, 28, 22, 12, 9, 6, 4, 2, 1, 8, 21, 31, 37, 40, 39, 35, 27, 16, 12, 9, 6, 4, 2, 1, 9, 24, 36, 44, 49, 50, 48, 42, 33, 20, 16

Offset

1,1

Comments

Row n consists of 2n+1 positive integers.

Links

Table of n, a(n) for n=1..74.

Example

First six rows:
  2   3   1
  3   6   6   2   1
  4   9  11   9   4   2   1
  5  12  16  16  13   6   4   2  1
  6  15  21  23  22  17   9   6  4  2  1
  7  18  26  30  31  28  22  12  9  6  4  2  1
For n=2, there are 6 triples (x,y,z) having x <= y:
  111:  |x-y| + |y-z| = 0
  112:  |x-y| + |y-z| = 1
  121:  |x-y| + |y-z| = 2
  122:  |x-y| + |y-z| = 1
  221:  |x-y| + |y-z| = 1
  222:  |x-y| + |y-z| = 0,
so row 1 of the array is (2,3,1), representing two 0s, three 1s, and one 1.

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, 0, 2 n - 2}];
v = Flatten[u]  (* sequence *)
Column[Table[Length[a[n, k]], {n, 2, 15}, {k, 0, 2 n - 2}]]  (* array *)

Crossrefs

Cf. A002411 (row sums), A002620 (limiting reverse row), A368434, A368437, A368515, A368516, A368517, A368519, A368520, A368521, A368522.

Sequence in context: A101912 A208522 A209569 * A176850 A208516 A111808

Adjacent sequences: A368155 A368156 A368157 * A368159 A368160 A368161

Keyword

nonn,tabf

Author

Clark Kimberling, Jan 20 2024

Status

approved