

A323131


Number of uncrossed rooted knight's paths of length n on an infinite board.


10



1, 7, 47, 303, 1921, 11963, 74130, 454484, 2779152, 16882278, 102384151, 618136584, 3727827148, 22408576099, 134595908277, 806452390868
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OFFSET

1,2


COMMENTS

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).


LINKS



EXAMPLE

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 selfavoiding 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.


CROSSREFS



KEYWORD

nonn,walk,more,hard


AUTHOR



EXTENSIONS



STATUS

approved



