login
Numerator of the square of the radius of the largest circle, centered at the origin, around which a Racetrack car (using von Neumann neighborhood) can run a full lap in n steps.
4

%I #8 Feb 19 2022 14:21:07

%S 1,1,2,2,4,9,9,9,16,32,32,196,81,125,392,1225,100,1681,160,4489,200,

%T 225,1369,320,400

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

%C The car starts and finishes on the positive x-axis, as in A351042.

%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="/A351351/a351351.svg">Some optimal Racetrack trajectories for A351351/A351352</a>.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Racetrack_(game)">Racetrack</a>

%F a(n)/A351352(n) <= A351349(n)/A351350(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 = 8 (r^2 = 1/2 = a(8)/A351352(8)):

%e . 3 1

%e 4 * 8

%e 5 7 .

%e .

%e n = 9 (r^2 = 1 = a(9)/A351352(9)):

%e . 3 2 . .

%e 4 . . 1 .

%e 5 . * 0 9

%e . 6 7 8 .

%e .

%e n = 10 (r^2 = 2 = a(10)/A351352(10)):

%e . . 3 2 .

%e . 4 . . 1

%e 5 . * . 10

%e 6 . . 9 .

%e . 7 8 . .

%e .

%e n = 12 (r^2 = 4 = a(12)/A351352(12)):

%e . 4 3 2 .

%e 5 . . . 1

%e 6 . * . 12

%e 7 . . . 11

%e . 8 9 10 .

%e .

%e n = 13 (r^2 = 9 = a(13)/A351352(13)):

%e . . . 4 . 3 . . . .

%e . 5 . . . . . 2 . .

%e 6 . . . . . . . 1 .

%e 7 . . . * . . . 0 13

%e 8 . . . . . . . . .

%e . 9 . . . . . 12 . .

%e . . . 10 . 11 . . . .

%Y Cf. A351042, A351349, A351350, A351352 (denominators).

%K nonn,frac,more

%O 8,3

%A _Pontus von Brömssen_, Feb 09 2022