A374438 Triangle read by rows: T(n, k) = T(n - 1, k) + T(n - 2, k - 2), with initial values T(n, k) = k + 1 for k < 3.

1, 1, 2, 1, 2, 3, 1, 2, 3, 2, 1, 2, 3, 4, 3, 1, 2, 3, 6, 6, 2, 1, 2, 3, 8, 9, 6, 3, 1, 2, 3, 10, 12, 12, 9, 2, 1, 2, 3, 12, 15, 20, 18, 8, 3, 1, 2, 3, 14, 18, 30, 30, 20, 12, 2, 1, 2, 3, 16, 21, 42, 45, 40, 30, 10, 3, 1, 2, 3, 18, 24, 56, 63, 70, 60, 30, 15, 2

Offset: 0

Author

Peter Luschny, Jul 22 2024

Keywords

• nonn
• tabl

Comments

See A374439 and the cross-references for comments about this family of triangles, where the recurrence is defined as in the name, but with an additional parameter m for the initial values: T(n, k) = k + 1 for k < m.

As m -> oo, the rows of the triangles become the initial segments of the integers.

Examples

			Triangle starts:
  [ 0] [1]
  [ 1] [1, 2]
  [ 2] [1, 2, 3]
  [ 3] [1, 2, 3,  2]
  [ 4] [1, 2, 3,  4,  3]
  [ 5] [1, 2, 3,  6,  6,  2]
  [ 6] [1, 2, 3,  8,  9,  6,  3]
  [ 7] [1, 2, 3, 10, 12, 12,  9,  2]
  [ 8] [1, 2, 3, 12, 15, 20, 18,  8,  3]
  [ 9] [1, 2, 3, 14, 18, 30, 30, 20, 12,  2]
  [10] [1, 2, 3, 16, 21, 42, 45, 40, 30, 10, 3]
		

Crossrefs

Family of triangles: A162515 (m=1, Fibonacci), A374439 (m=2, Lucas), this triangle (m=3).

Row sums: A187890 (apart from initial terms), also A001060 + 1 (with 1 prepended).

Cf. A006355 (odd sums), A187893 (even sums).

Cf. related to deltas: A065220, A210673.

Programs

Maple

M := 3;  # family index
T := proc(n, k) option remember; if k > n then 0 elif k < M then k + 1 else
T(n - 1, k) + T(n - 2, k - 2) fi end:
seq(seq(T(n, k), k = 0..n), n = 0..11);

Python

from functools import cache
@cache
def T(n: int, k: int) -> int:
    if k > n: return 0
    if k < 3: return k + 1
    return T(n - 1, k) + T(n - 2, k - 2)