%I #14 Feb 19 2022 14:20:48
%S 1,1,1,4,4,81,9,16,16,576,36,36,64,81,1250,100,144,144,8100,225
%N 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 The car starts and finishes on the positive x-axis, as in A351041.
%C 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 Pontus von Brömssen, <a href="/A351349/a351349.svg">Some optimal Racetrack trajectories for A351349/A351350</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Racetrack_(game)">Racetrack</a>
%F a(n)/A351350(n) >= A351351(n)/A351352(n).
%e 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 .
%e n = 6 (r^2 = 1/2 = a(6)/A351350(6)):
%e . 1 .
%e 3 * 6
%e 4 5 .
%e .
%e n = 7 (r^2 = 1 = a(7)/A351350(7)):
%e . 2 . 1 .
%e 3 . * . 7
%e . 5 . 6 .
%e .
%e n = 9 (r^2 = 4 = a(9)/A351350(9)):
%e . 3 . 2 .
%e 4 . . . 1
%e . . * . 9
%e 5 . . . 8
%e . 6 . 7 .
%e .
%e n = 11 (r^2 = 81/10 = a(11)/A351350(11)):
%e . 4 . 3 . . . . . .
%e 5 . . . . . 2 . . .
%e . . . . . . . . 1 .
%e 6 . . * . . . . 11 0
%e . . . . . . . . . .
%e 7 . . . . . 10 . . .
%e . 8 . 9 . . . . . .
%e .
%e n = 12 (r^2 = 9 = a(12)/A351350(12)):
%e . . . 4 . 3 . . .
%e . 5 . . . . . 2 .
%e . . . . . . . . 1
%e 6 . . . * . . . 12
%e 7 . . . . . . . .
%e . 8 . . . . . 11 .
%e . . . 9 . 10 . . .
%Y Cf. A351041, A351350 (denominators), A351351, A351352.
%K nonn,frac,more
%O 6,4
%A _Pontus von Brömssen_, Feb 09 2022