A380812 Sequence of x-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, 1, 2, 2, 1, 0, -1, -1, -1, -1, 0, 1, 2, 2, 2, 1, 0, -1, -2, -3, -3, -2, -1, 0, 1, 3, 4, 4, 4, 3, 2, 1, -1, -3, -4, -4, -4, -4, -3, -2, 0, 2, 4, 5, 5, 4, 3, 2, 0, -2, -4, -5, -5, -5, -5, -4, -3, -1, 1, 3, 4, 4, 3, 2, 1, -1, -3, -5, -6, -6, -5, -3, 0, 3, 6, 8

Offset: 0

Author

Pontus von Brömssen, Feb 05 2025

Keywords

• sign

Comments

The car starts at the origin and thereafter moves, according to the rules of Racetrack with von Neumann neighborhood (see A351042 or Wikipedia link), to the unvisited square that has the lowest spiral number, provided that it is possible to extend the trajectory to an infinite one. The spiral numbering is described in A316328.

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.

Examples

			In the 144th step, the car moves from (-9,-8) to (-6,-6) (a(144) = A380813(144) = -6). A priori, the next possible positions (ordered by increasing spiral number) are (-3,-3), (-4,-4), (-3,-4), (-2,-4), and (-3,-5). Of these, (-3,-3) has already been visited (after the 103rd step), so the next choice is (-4,-4). From that position, however, the car is forced to move to (-2,-2) (all other alternatives have already been visited), and from (-2,-2) there are no available positions not already visited (so the trajectory in A351043 ends there). The next option (-3,-4) is also a dead end, but from (-2,-4) it is possible to continue forever, so a(145) = -2 and A380813(145) = -4.
		

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.
• Pontus von Brömssen, Plot of trajectory, using Plot2.
• Wikipedia, Racetrack.

Crossrefs

Cf. A174344, A316328, A351042, A351043, A380813 (y-coordinates), A380814.

Formula

a(n) = A174344(A351043(n)+1) for n <= 144.