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Minimal number of steps for a Racetrack car (using von Neumann neighborhood) to go around a circle of radius n.
8

%I #20 Feb 13 2022 09:23:29

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

%T 36,37

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

%C 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.

%H Pontus von Brömssen, <a href="/A351042/a351042.svg">Examples of optimal trajectories in A351042 for 1 <= n <= 8</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Racetrack_(game)">Racetrack</a>

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

%F a(n) >= A351041(n).

%e The following diagrams show examples of optimal trajectories for n = 1, 2, 3. The origin is marked with an asterisk.

%e .

%e a(1) = 9:

%e . 3 2 . .

%e 4 . . 1 .

%e 5 . * 0 9

%e . 6 7 8 .

%e .

%e a(2) = 12:

%e . 4 3 2 . .

%e 5 . . . 1 .

%e 6 . * . 0 12

%e 7 . . . 11 .

%e . 8 9 10 . .

%e .

%e a(3) = 13:

%e . . . 4 . 3 . . . .

%e . 5 . . . . . 2 . .

%e 6 . . . . . . . 1 .

%e 7 . . . * . . . 0 13

%e 8 . . . . . . . . .

%e . 9 . . . . . 12 . .

%e . . . 10 . 11 . . . .

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

%K nonn,more

%O 1,1

%A _Pontus von Brömssen_, Jan 30 2022