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Triangle read by rows: T(m,n) is the number of paths for a Racetrack car (using Moore neighborhood) with initial velocity zero, going from one corner to the diagonally opposite corner on an m X n grid, such that all positions are visited exactly once, 1 <= n <= m.
2

%I #11 Feb 18 2023 08:07:37

%S 1,1,0,1,1,6,1,0,15,2,1,1,70,289,9436,1,0,294,191,128020

%N Triangle read by rows: T(m,n) is the number of paths for a Racetrack car (using Moore neighborhood) with initial velocity zero, going from one corner to the diagonally opposite corner on an m X n grid, such that all positions are visited exactly once, 1 <= n <= m.

%C For a Racetrack car using von Neumann neighborhood (see A351042), there are no such paths if 2 <= n <= m, because the car will never be able to leave a corner of the grid (except the corner where it starts).

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

%e Triangle begins:

%e m\n| 1 2 3 4 5 6

%e ---+-----------------------

%e 1 | 1

%e 2 | 1 0

%e 3 | 1 1 6

%e 4 | 1 0 15 2

%e 5 | 1 1 70 289 9436

%e 6 | 1 0 294 191 128020 ?

%Y Cf. A000012 (column n=1), A000035 (column n=2), A272445, A351041, A351042, A351106, A351111 (main diagonal).

%K nonn,tabl,more

%O 1,6

%A _Pontus von Brömssen_, Feb 01 2022