A368606 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, 2, 2, 1, 3, 4, 4, 2, 1, 4, 6, 7, 6, 4, 2, 1, 5, 8, 10, 10, 9, 6, 4, 2, 1, 6, 10, 13, 14, 14, 12, 9, 6, 4, 2, 1, 7, 12, 16, 18, 19, 18, 16, 12, 9, 6, 4, 2, 1, 8, 14, 19, 22, 24, 24, 23, 20, 16, 12, 9, 6, 4, 2, 1, 9, 16, 22, 26, 29, 30, 30, 28, 25, 20, 16

Offset

1,2

Comments

Row n consists of 2n-1 positive integers.

Links

Table of n, a(n) for n=1..75.

Example

First six rows:
  1
  2    2    1
  3    4    4    2    1
  4    6    7    6    4    2   1
  5    8   10   10    9    6   4   2   1
  6   10   13   14   14   12   9   6   4   2   1
For n=2, there are 5 triples (x,y,z) having x <= y and y >= z:
  111:  |x-y| + |y-z| = 0
  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 2 of the array is (2,2,1), representing two 0s, two 1s, and one 3.

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, 1, 15}, {k, 0, 2 n - 2}];
v = Flatten[u]  (* sequence *)
Column[Table[Length[a[n, k]], {n, 1, 15}, {k, 0, 2 n - 2}]]  (* array *)

Crossrefs

Cf. A000027 (column 1), A000330 (row sums), A002620 (limiting reversed row), A368434, A368437, A368515, A368516, A368517, A368518, A368519, A368520, A368521, A368522, A368604, A368605, A368607, A368609.

Sequence in context: A193820 A216368 A123956 * A331953 A113594 A368604

Adjacent sequences: A368603 A368604 A368605 * A368607 A368608 A368609

Keyword

nonn,tabf

Author

Clark Kimberling, Jan 22 2024

Status

approved