A351043 -id:A351043 - cp's OEIS frontend

A351042 Minimal number of steps for a Racetrack car (using von Neumann neighborhood) to go around a circle of radius n.

9, 12, 13, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 32, 32, 34, 34, 36, 36, 37

Offset: 1

Pontus von Brömssen, Jan 30 2022

The car moves according to the rules of the game of Racetrack with von Neumann neighborhood, i.e., if P, Q, and R are three successive positions of the car, one coordinate of the second difference (acceleration vector) P - 2Q + R must be 0, and the other 1, 0, or -1. The car starts with zero velocity at a point (x,0) for some integer x >= n, and finishes when it passes, or lands on, the positive x-axis after a complete counterclockwise lap around the origin. The line segments between successive positions must be outside or on the circle with center in (0,0) and radius n.

			The following diagrams show examples of optimal trajectories for n = 1, 2, 3. The origin is marked with an asterisk.
.
  a(1) = 9:
  .  3  2  .  .
  4  .  .  1  .
  5  .  *  0  9
  .  6  7  8  .
.
  a(2) = 12:
  .  4  3  2  .  .
  5  .  .  .  1  .
  6  .  *  .  0 12
  7  .  .  . 11  .
  .  8  9 10  .  .
.
  a(3) = 13:
  .  .  .  4  .  3  .  .  .  .
  .  5  .  .  .  .  .  2  .  .
  6  .  .  .  .  .  .  .  1  .
  7  .  .  .  *  .  .  .  0 13
  8  .  .  .  .  .  .  .  .  .
  .  9  .  .  .  .  . 12  .  .
  .  .  . 10  . 11  .  .  .  .

Pontus von Brömssen, Examples of optimal trajectories in A351042 for 1 <= n <= 8.
Wikipedia, Racetrack

Cf. A027434, A351041, A351043, A351351, A351352.

a(n) = min {k >= 8; A351351(k)/A351352(k) >= n^2}.

a(n) >= A351041(n).

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.

Original entry on oeis.org

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

Pontus von Brömssen, Feb 05 2025

sign

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.

			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.

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.

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

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

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.

Original entry on oeis.org

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

Pontus von Brömssen, Feb 05 2025

sign

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.

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.

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

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

cp's OEIS Frontend

A351042 Minimal number of steps for a Racetrack car (using von Neumann neighborhood) to go around a circle of radius n.

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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.

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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.

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Author

Keywords

Comments

Links

Crossrefs

Formula