A372421
Number of steps required to kill the hydra in a version of the hydra game (see comments) where the rightmost head is chopped off in each step and new heads are grown to the left.
Original entry on oeis.org
0, 1, 3, 9, 49, 1230, 757071, 286578628063, 41063655031378934880024, 843111882268046256673111236649909091104560309, 355418823010783945962646271385485944012152784388172734299894340514265378207290093661367905
Offset: 0
For n = 3, the first three steps are illustrated in the diagrams below. In these diagrams, "R" denotes the root, "o" internal nodes, "X" the head to be chopped off, and "H" other heads.
.
H H H H H H
/ |/ \|/
R--o--o--X => R--o--X => R--o--X => H--R--X
/
H
.
After this no more heads will grow, so another 6 steps are needed to chop off the remaining heads. The total number of steps is thus a(3) = 3 + 6 = 9.
Last element in each row of
A372593.
A180368
Number of steps of the one-sided hydra process for a linear tree of length n.
Original entry on oeis.org
0, 1, 3, 8, 38, 161915
Offset: 0
Here is the sequence of hydra transformations for a(3) = 8.
Sequence of heights is 3,2,2,2,2,2,1,1,0.
Sequence of node counts is 4,4,5,5,4,3,3,2,1.
Sequence of head counts is 1,2,2,3,2,1,2,1,0.
x is the head that will be cut off at the next step:
x
|
o x o x o o o x
| |/ | | | | |
o o o o x o o x o o x o x
| | |/ \|/ |/ | |/ |
o => o => o => o => o => o => o => o => o
Last element in each row of
A372594.
-
b:= proc(h) local f; f:= h[1];
subsop(1=`if`(f=[], NULL,
`if`(f[1]=[], (subsop(1=NULL, f))$2
, b(f))), h)
end:
a:= proc(n) local i, t;
[];
for i to n do [%] od;
for t from 0 while %<>[] do b(%) od; t
end:
seq(a(n), n=0..5); # Alois P. Heinz, Mar 31 2011
A372592
Irregular triangle read by rows, where the n-th row gives the number of steps in the hydra game when the initial hydra is each of the A000108(n) ordered trees with n edges (ordered by lexicographic order of their corresponding Dyck words as in A063171) and new heads are grown to the right.
Original entry on oeis.org
0, 1, 2, 3, 3, 4, 5, 7, 11, 4, 5, 6, 8, 12, 7, 9, 11, 15, 23, 31, 79, 447, 1114111, 5, 6, 7, 9, 13, 8, 10, 12, 16, 24, 32, 80, 448, 1114112, 9, 11, 13, 17, 25, 15, 19, 23, 31, 47, 63, 159, 895, 2228223, 79, 191, 447, 2303, 53247, 1114111, 45079976738815, 6065988000108893953800078394579416901568357495071628808248312306073599
Offset: 0
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.
Last elements on each row give
A372101.
A372478
Number of steps required to kill a Kirby-Paris hydra composed of a linear graph with n edges where, after removing the rightmost head at step s, s new subtrees sprout from the head's grandparent node (see comments).
Original entry on oeis.org
In the following tree diagrams R is the root, o is a node and H is a head (leaf). Head chopping (leaf removal) is denoted by X.
For n = 2, the sequence of the 3 choppings is:
.
H X
\ \
o o H H X X
\ \ / \ / \
R R R R
.
For n = 3, the sequence of the 37 choppings is:
.
H X
\ \
o o H H X H H H H X H H
\ \ / \| | / \ | / \ |
o o o o o o o o H H H o o X X X X
\ \ \|/ \|/ / / / \|/ / / /
R R R R------ R------
.
H X H X
\ | \ \
o o H (8) H o X (9) X o H (18) H X (19) X
\|/ ... / \ / ... / \ / ... / \ ... /
R------ R------ R--------- ---R---
.
Last element in each row of
A372595.
Showing 1-4 of 4 results.
Comments