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Number of simple paths for a Racetrack car (using Moore neighborhood) with initial velocity zero, going from one corner to the diagonally opposite corner on an n X n grid.
3

%I #7 Feb 04 2022 14:40:23

%S 1,3,23,1470,914525

%N Number of simple paths for a Racetrack car (using Moore neighborhood) with initial velocity zero, going from one corner to the diagonally opposite corner on an n X n grid.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Racetrack_(game)">Racetrack</a>

%e For n = 3 the following paths exist (up to reflection in the diagonal). The numbers give the positions of the car after successive steps.

%e ..2 ..3 ..3 ..3 ..4 ..4 .34 .56 456 548 678 678

%e .1. ..2 .2. .12 ..3 .23 .2. .43 32. 673 543 512

%e 0.. 01. 01. 0.. 012 01. 01. 012 01. 012 012 043

%e Of these, only the first path is symmetric with respect to the diagonal, so the other 11 give rise to 2 paths each. In total, there are a(3) = 1 + 2*11 = 23 possible paths.

%Y Main diagonal of A351106.

%Y Cf. A140518, A351041, A351109, A351111.

%K nonn,more

%O 1,2

%A _Pontus von Brömssen_, Feb 01 2022