OFFSET
0,2
COMMENTS
The paths through the tree represent the compositions counted in A173258 that have first part 1.
For rows n > 1, row n starts with row n-2.
Any positive number k will first appear in the (k-1)-th row and thereafter in rows of opposite parity to k. The number of times k will appear in row n is A053121(n,k-1).
Row n >= 1 gives the row lengths of the Christmas tree pattern of order n (cf. A367508). - Paolo Xausa, Nov 26 2023
A new row can be generated by applying the morphism 1 -> 2, t -> {t-1,t+1} (for t > 1) to the previous row. - Paolo Xausa, Dec 08 2023
LINKS
Paolo Xausa, Table of n, a(n) for n = 0..13494 (rows 0..15 of the triangle, flattened).
EXAMPLE
Triangle begins:
n=0: 1;
n=1: 2;
n=2: 1, 3;
n=3: 2, 2, 4;
n=4: 1, 3, 1, 3, 3, 5;
n=5: 2, 2, 4, 2, 2, 4, 2, 4, 4, 6;
n=6: 1, 3, 1, 3, 3, 5, 1, 3, 1, 3, 3, 5, 1, 3, 3, 5, 3, 5, 5, 7;
...
The binary tree starts with root 1 in row n = 0. In row n = 2, the parent node 2 has the first left child since 2 - 1 > 0.
The tree begins:
row
[n]
[0] 1
\
[1] _________2_________
/ \
[2] 1 _____3_____
\ / \
[3] __2__ __2__ __4__
/ \ / \ / \
[4] 1 3 1 3 3 5
\ / \ \ / \ / \ / \
[5] 2 2 4 2 2 4 2 4 4 6
.
MATHEMATICA
SubstitutionSystem[{1->{2}, t_/; t>1:>{t-1, t+1}}, {1}, 8] (* Paolo Xausa, Dec 23 2023 *)
PROG
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
John Tyler Rascoe, Jun 17 2023
STATUS
approved