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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 (list; graph; refs; listen; history; text; internal format)
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).
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.
Sequence in context: A009260 A201871 A198845 * A229010 A126635 A085352
Hugo Pfoertner, Jan 05 2019
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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Last modified May 21 11:59 EDT 2024. Contains 372736 sequences. (Running on oeis4.)