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A323562
Number of rooted self-avoiding king's walks on an infinite chessboard trapped after n moves.
9
8, 200, 2446, 21946, 169782, 1205428, 8119338, 52862872, 336465352, 2108185746
OFFSET
8,1
COMMENTS
The first step is either (0,0)->(1,0) or (0,0)->(1,1). Rotated paths are not counted separately.
The average number of moves of a self-avoiding random walk of a king on an infinite chessboard to self-trapping is 209.71. The corresponding number of moves for paths with forbidden crossing (A323141) is 69.865.
a(n)=0 for n<8.
EXAMPLE
a(8) = 8, because the following 8 walks of 8 moves of a king starting at S with a first move (0,0)->(1,0) visit all neighbors of the trapping location T. The starting point itself is also blocked. There are no such shortest walks with first move (0,0)->(1,1).
.
o <-- o <-- o o o <-- o o --> o --> o o <-- o <-- o
| ^ ^ \ / ^ ^ | | ^
v | | / \ | | v v |
o --> T o o T o o T o o T o
^ ^ \ \ | | / ^
| | \ \ v v / |
S --> o --> o S --> o --> o S --> o o o S --> o
.
S --> o --> o S --> o --> o S --> o o o S --> o
| | / / ^ ^ \ |
v v / / | | \ v
o --> T o o T o o T o o T o
^ | | \ / | | ^ ^ |
| v v / \ v v | | v
o <-- o <-- o o o <-- o o --> o --> o o <-- o <-- o
- Hugo Pfoertner, Jul 23 2020
CROSSREFS
KEYWORD
nonn,walk,more
AUTHOR
Hugo Pfoertner, Jan 17 2019
STATUS
approved