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A323131
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Number of uncrossed rooted knight's paths of length n on an infinite board.
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9
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1, 7, 47, 303, 1921, 11963, 74130, 454484, 2779152, 16882278, 102384151, 618136584, 3727827148, 22408576099, 134595908277, 806452390868
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OFFSET
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1,2
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COMMENTS
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The direction of the first move is kept fixed.
The average number of steps of a random walk using such knight moves with forbidden crossing is 45 (compare to A322831).
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LINKS
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Table of n, a(n) for n=1..16.
Hugo Pfoertner, Illustrations of rooted uncrossed knight's paths of length <= 3, (2019).
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EXAMPLE
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a(1) = 1: The fixed initial move.
a(2) = 7: Relative to the direction given by the initial move, there are 7 possible direction changes. The backward direction is illegal for the self-avoiding uncrossed path. Only for the right angle turn its mirror image would coincide with the turn in the opposite direction. Therefore this move would be eliminated in the unrooted walks, making a(2) > A323132(2) = 6.
a(3) = 47: 2 of all 7*7 = 49 continuation moves already lead to a crossing of the first path segment.
See illustrations at Pfoertner link.
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CROSSREFS
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Cf. A003192, A272773, A323132, A323133, A323134, A323559.
Sequence in context: A009260 A201871 A198845 * A229010 A126635 A085352
Adjacent sequences: A323128 A323129 A323130 * A323132 A323133 A323134
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KEYWORD
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nonn,walk,more,hard
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AUTHOR
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Hugo Pfoertner, Jan 05 2019
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EXTENSIONS
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Erroneous (as pointed out by Bert Dobbelaere) a(8) and a(10) corrected by Hugo Pfoertner, Jan 18 2019
a(12)-a(16) from Bert Dobbelaere, Jan 18 2019
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STATUS
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approved
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