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A323562
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Number of rooted self-avoiding king's walks on an infinite chessboard trapped after n moves.
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7
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OFFSET
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8,1
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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).
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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
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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
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CROSSREFS
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KEYWORD
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nonn,walk,more
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AUTHOR
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STATUS
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approved
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