|
%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
|
