OFFSET
0,3
COMMENTS
As in A372101, the rightmost head (leaf) is always chopped off, and after the m-th head is chopped off (if it is not directly connected to the root) m new heads grow from the node two levels closer to the root from the head chopped off (its grandparent) to the right of all existing branches of that node.
T(5,37) = 20472...84351 (167697 digits). The corresponding initial tree is represented by the bracket string "((()(())))" (the 37th Dyck word on 5 pairs of brackets).
FORMULA
T(n,k) = T(n-1,k)+1 if 1 <= k <= A000108(n-1).
EXAMPLE
Triangle begins:
0;
1;
2, 3;
3, 4, 5, 7, 11;
4, 5, 6, 8, 12, 7, 9, 11, 15, 23, 31, 79, 447, 1114111;
...
For n = 4, k = 10, the hydra game for the initial tree corresponding to the bracket string "(()(()))" (the 10th Dyck word on 4 pairs of brackets) is shown below. The root is denoted by "R", internal nodes by "o", the head to be chopped off by "X", other heads by "H". Numbers below the arrows show how many steps that are required to go from the tree on the left to the tree on the right.
.
X
/
H o H H H H H X X
\ / \ / \ / \ / \
o o--X o H o o X
| | |/ | | |
R => R => R--X => R => R => R => R
T(4,10) = 1 + 1 + 2 + 6 + 12 + 1 = 23.
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Pontus von Brömssen, May 06 2024
STATUS
approved