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

%I A351042 #20 Feb 13 2022 09:23:29
%S A351042 9,12,13,16,17,19,21,22,23,24,25,26,27,28,29,30,31,32,32,32,34,34,36,
%T A351042 36,37
%N A351042 Minimal number of steps for a Racetrack car (using von Neumann neighborhood) to go around a circle of radius n.
%C A351042 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 A351042 Pontus von Brömssen, <a href="/A351042/a351042.svg">Examples of optimal trajectories in A351042 for 1 <= n <= 8</a>.
%H A351042 Wikipedia, <a href="https://en.wikipedia.org/wiki/Racetrack_(game)">Racetrack</a>
%F A351042 a(n) = min {k >= 8; A351351(k)/A351352(k) >= n^2}.
%F A351042 a(n) >= A351041(n).
%e A351042 The following diagrams show examples of optimal trajectories for n = 1, 2, 3. The origin is marked with an asterisk.
%e A351042 .
%e A351042   a(1) = 9:
%e A351042   .  3  2  .  .
%e A351042   4  .  .  1  .
%e A351042   5  .  *  0  9
%e A351042   .  6  7  8  .
%e A351042 .
%e A351042   a(2) = 12:
%e A351042   .  4  3  2  .  .
%e A351042   5  .  .  .  1  .
%e A351042   6  .  *  .  0 12
%e A351042   7  .  .  . 11  .
%e A351042   .  8  9 10  .  .
%e A351042 .
%e A351042   a(3) = 13:
%e A351042   .  .  .  4  .  3  .  .  .  .
%e A351042   .  5  .  .  .  .  .  2  .  .
%e A351042   6  .  .  .  .  .  .  .  1  .
%e A351042   7  .  .  .  *  .  .  .  0 13
%e A351042   8  .  .  .  .  .  .  .  .  .
%e A351042   .  9  .  .  .  .  . 12  .  .
%e A351042   .  .  . 10  . 11  .  .  .  .
%Y A351042 Cf. A027434, A351041, A351043, A351351, A351352.
%K A351042 nonn,more
%O A351042 1,1
%A A351042 _Pontus von Brömssen_, Jan 30 2022