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A351349 Numerator of the square of the radius of the largest circle, centered at the origin, around which a Racetrack car (using Moore neighborhood) can run a full lap in n steps.

%I A351349 #14 Feb 19 2022 14:20:48
%S A351349 1,1,1,4,4,81,9,16,16,576,36,36,64,81,1250,100,144,144,8100,225
%N A351349 Numerator of the square of the radius of the largest circle, centered at the origin, around which a Racetrack car (using Moore neighborhood) can run a full lap in n steps.
%C A351349 The car starts and finishes on the positive x-axis, as in A351041.
%C A351349 The square of the radius of the largest circle is a rational number, because the squared distance from the origin to a line segment between two points with integer coordinates is always rational.
%H A351349 Pontus von Brömssen, <a href="/A351349/a351349.svg">Some optimal Racetrack trajectories for A351349/A351350</a>.
%H A351349 Wikipedia, <a href="https://en.wikipedia.org/wiki/Racetrack_(game)">Racetrack</a>
%F A351349 a(n)/A351350(n) >= A351351(n)/A351352(n).
%e A351349 The following diagrams show examples of optimal trajectories for some values of n. The position of the car after k steps is labeled with the number k. If a number is missing, it means that the car stands still on that step. If the number 0 is missing (for the starting position), it means that the starting and finishing positions coincide. The origin is marked with an asterisk.
%e A351349 .
%e A351349   n = 6 (r^2 = 1/2 = a(6)/A351350(6)):
%e A351349   .  1  .
%e A351349   3  *  6
%e A351349   4  5  .
%e A351349 .
%e A351349   n = 7 (r^2 = 1 = a(7)/A351350(7)):
%e A351349   .  2  .  1  .
%e A351349   3  .  *  .  7
%e A351349   .  5  .  6  .
%e A351349 .
%e A351349   n = 9 (r^2 = 4 = a(9)/A351350(9)):
%e A351349   .  3  .  2  .
%e A351349   4  .  .  .  1
%e A351349   .  .  *  .  9
%e A351349   5  .  .  .  8
%e A351349   .  6  .  7  .
%e A351349 .
%e A351349   n = 11 (r^2 = 81/10 = a(11)/A351350(11)):
%e A351349   .  4  .  3  .  .  .  .  .  .
%e A351349   5  .  .  .  .  .  2  .  .  .
%e A351349   .  .  .  .  .  .  .  .  1  .
%e A351349   6  .  .  *  .  .  .  . 11  0
%e A351349   .  .  .  .  .  .  .  .  .  .
%e A351349   7  .  .  .  .  . 10  .  .  .
%e A351349   .  8  .  9  .  .  .  .  .  .
%e A351349 .
%e A351349   n = 12 (r^2 = 9 = a(12)/A351350(12)):
%e A351349   .  .  .  4  .  3  .  .  .
%e A351349   .  5  .  .  .  .  .  2  .
%e A351349   .  .  .  .  .  .  .  .  1
%e A351349   6  .  .  .  *  .  .  . 12
%e A351349   7  .  .  .  .  .  .  .  .
%e A351349   .  8  .  .  .  .  . 11  .
%e A351349   .  .  .  9  . 10  .  .  .
%Y A351349 Cf. A351041, A351350 (denominators), A351351, A351352.
%K A351349 nonn,frac,more
%O A351349 6,4
%A A351349 _Pontus von Brömssen_, Feb 09 2022