cp's OEIS Frontend

A351041 Minimal number of steps for a Racetrack car (using Moore neighborhood) to go around a circle of radius n.

%I A351041 #22 Feb 13 2022 09:23:34
%S A351041 7,9,12,13,15,16,18,18,19,21,22,22,24,24,25,26,27,27,28,28,30,31,31,
%T A351041 31,32,33,33,34,35,35,36,36,37,37,37
%N A351041 Minimal number of steps for a Racetrack car (using Moore neighborhood) to go around a circle of radius n.
%C A351041 The car moves according to the rules of the game of Racetrack, i.e., if P, Q, and R are three successive positions of the car, both coordinates of the second difference (acceleration vector) P - 2Q + R must be 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 A351041 Pontus von Brömssen, <a href="/A351041/a351041.svg">Examples of optimal trajectories in A351041 for 1 <= n <= 8</a>.
%H A351041 Wikipedia, <a href="https://en.wikipedia.org/wiki/Racetrack_(game)">Racetrack</a>
%F A351041 a(n) = min {k >= 6; A351349(k)/A351350(k) >= n^2}.
%F A351041 a(n) <= A351042(n).
%F A351041 a(n) >= A027434(n) + A027434(2*n) + A002024(n). This can be seen by looking at the y-coordinate only: First, the car must go up to at least y = n and reduce the speed in the y-direction to zero in order to turn downwards; this requires at least A027434(n) steps. Then down to y = -n or below with speed reduced to zero; this requires at least A027434(2*n) steps. Finally, up to at least y = 0 (not necessarily reducing the speed); this requires at least A002024(n) steps.
%F A351041 It appears that a(n) = A027434(n) + A027434(2*n) + A002024(n) + 1 if n is a triangular number (A000217), otherwise a(n) = A027434(n) + A027434(2*n) + A002024(n).
%e A351041 The following diagrams show examples of optimal trajectories for n = 1, 2, 3. The origin is marked with an asterisk.
%e A351041 .
%e A351041   a(1) = 7:
%e A351041   .  2  .  1  .  .
%e A351041   3  .  *  .  0  7
%e A351041   .  5  .  6  .  .
%e A351041   (The car stands still on the fourth step.)
%e A351041 .
%e A351041   a(2) = 9:
%e A351041   .  3  .  2  .  .
%e A351041   4  .  .  .  1  .
%e A351041   .  .  *  .  0  9
%e A351041   5  .  .  .  8  .
%e A351041   .  6  .  7  .  .
%e A351041 .
%e A351041   a(3) = 12:
%e A351041   .  .  .  4  3  .  .  .  .
%e A351041   .  5  .  .  .  .  2  .  .
%e A351041   .  .  .  .  .  .  .  .  .
%e A351041   6  .  .  .  .  .  .  1  .
%e A351041   7  .  .  .  *  .  .  0 12
%e A351041   .  .  .  .  .  .  .  .  .
%e A351041   .  8  .  .  .  .  . 11  .
%e A351041   .  .  .  9  . 10  .  .  .
%Y A351041 Cf. A000217, A002024, A027434, A351042, A351349, A351350.
%K A351041 nonn,more
%O A351041 1,1
%A A351041 _Pontus von Brömssen_, Jan 29 2022