A351111 -id:A351111 - cp's OEIS frontend

A351107 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.

1, 3, 23, 1470, 914525

Offset: 1

Pontus von Brömssen, Feb 01 2022

			For n = 3 the following paths exist (up to reflection in the diagonal). The numbers give the positions of the car after successive steps.
  ..2  ..3  ..3  ..3  ..4  ..4  .34  .56  456  548  678  678
  .1.  ..2  .2.  .12  ..3  .23  .2.  .43  32.  673  543  512
  0..  01.  01.  0..  012  01.  01.  012  01.  012  012  043
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.

Wikipedia, Racetrack

Main diagonal of A351106.

Cf. A140518, A351041, A351109, A351111.

A351110 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.

Original entry on oeis.org

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

Offset: 1

Pontus von Brömssen, Feb 01 2022

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).

			Triangle begins:
  m\n| 1  2   3   4      5  6
  ---+-----------------------
  1  | 1
  2  | 1  0
  3  | 1  1   6
  4  | 1  0  15   2
  5  | 1  1  70 289   9436
  6  | 1  0 294 191 128020  ?

Wikipedia, Racetrack

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

cp's OEIS Frontend

A351107 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.

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