A323131 Number of uncrossed rooted knight's paths of length n on an infinite board.

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

Offset: 1

Author

Hugo Pfoertner, Jan 05 2019

Keywords

• nonn
• walk
• more
• hard

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).

Examples

			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.
		

Links

• Hugo Pfoertner, Illustrations of rooted uncrossed knight's paths of length <= 3, (2019).

Crossrefs

Cf. A003192, A272773, A323132, A323133, A323134, A323559.

Extensions

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