A337277 Stern's triangle read by rows.

1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 2, 1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1, 1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 5, 4, 7, 3, 8, 5, 7, 2, 7, 5, 8, 3, 7, 4, 5, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1

Offset: 0

Author

N. J. A. Sloane, Sep 09 2020

Keywords

• nonn
• tabf

Comments

The first two rows are 1, then 1,1,1. To get row n, copy row n-1, and insert c+d between every pair of adjacent terms c,d, and finally insert a 1 at the beginning and end of the row.

The maximum value in row n is A000045(n+1). - Alois P. Heinz, Sep 09 2020

Examples

			Triangle begins:
  1;
  1, 1, 1;
  1, 1, 2, 1, 2, 1, 1;
  1, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 2, 1, 1;
  1, 1, 2, 1, 3, 2, 3, 1, 4, 3, 5, 2, 5, 3, 4, 1, 4, 3, 5, 2, 5, 3, 4, 1, 3, 2, 3, 1, 2, 1, 1;
  ...
		

References

• Stanley, Richard P. "Some Linear Recurrences Motivated by Stern’s Diatomic Array." The American Mathematical Monthly 127.2 (2020): 99-111.

Links

• Alois P. Heinz, Rows n = 0..14, flattened
• Richard P. Stanley, Some Linear Recurrences Motivated by Stern's Diatomic Array, arXiv:1901.04647 [math.CO], 2019.

Crossrefs

Cf. A000045, A002487, A049456, A334627.

Row sums give A000244.

Row lengths give A126646.

Programs

Maple

T:= proc(n) option remember; `if`(n=0, 1, (L-> [1, L[1], seq(
      [L[i-1]+L[i], L[i]][], i=2..nops(L)), 1][])([T(n-1)]))
    end:
seq(T(n), n=0..6);  # Alois P. Heinz, Sep 09 2020

Mathematica

Nest[Append[#, Flatten@ Join[{1}, If[Length@ # > 1, Map[{#1, #1 + #2} & @@ # &, Partition[#[[-1]], 2, 1] ], {}], {#[[-1, -1]]}, {1}]] &, {{1}}, 5] // Flatten (* Michael De Vlieger, Sep 09 2020 *)

Formula

T(n,n) = A002487(n+1). - Alois P. Heinz, Sep 09 2020