A380813 Sequence of y-coordinates of the lexicographically earliest (according to the spiral numbering of the square grid; see comments) infinite Racetrack trajectory (using von Neumann neighborhood) on the square grid.

0, 0, -1, -2, -3, -3, -2, -1, 1, 2, 3, 3, 2, 1, 0, -1, -1, 0, 2, 3, 4, 5, 5, 4, 2, 0, -2, -4, -5, -6, -6, -5, -4, -2, 0, 2, 4, 5, 6, 6, 6, 5, 3, 1, -1, -3, -4, -4, -4, -3, -1, 1, 3, 5, 6, 7, 7, 7, 6, 4, 2, 0, -2, -3, -4, -5, -5, -4, -3, -2, -1, 0, 1, 1, 0, -1

Offset: 0

Author

Pontus von Brömssen, Feb 05 2025

Keywords

• sign

Comments

See A380812 for more details.

See A351042 or Wikipedia link for a description of the rules of Racetrack.

The trajectory in A351043 is defined in a similar way, but it does not backtrack when it gets stuck, so it is finite, ending after 146 steps. The trajectory here is identical to the trajectory in A351043 for the first 144 steps, and it turns out that the y-coordinates agree also for the last 2 steps.

Links

• Pontus von Brömssen, Table of n, a(n) for n = 0..10000
• Pontus von Brömssen, Plot of the first 5000 steps of the trajectory.
• Wikipedia, Racetrack.

Crossrefs

Cf. A274923, A351042, A351043, A380812 (x-coordinates), A380814.

Formula

a(n) = A274923(A351043(n)+1) for n <= 146.