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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.
4

%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