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

Author

Pontus von Brömssen, Jan 30 2022

Keywords

• nonn
• more

Comments

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.

Examples

			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  .  .  .  .
		

Links

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

Crossrefs

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

Formula

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

a(n) >= A351041(n).