cp's OEIS Frontend

A351351 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.

%I A351351 #8 Feb 19 2022 14:21:07
%S A351351 1,1,2,2,4,9,9,9,16,32,32,196,81,125,392,1225,100,1681,160,4489,200,
%T A351351 225,1369,320,400
%N A351351 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 A351351 The car starts and finishes on the positive x-axis, as in A351042.
%C A351351 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 A351351 Pontus von Brömssen, <a href="/A351351/a351351.svg">Some optimal Racetrack trajectories for A351351/A351352</a>.
%H A351351 Wikipedia, <a href="https://en.wikipedia.org/wiki/Racetrack_(game)">Racetrack</a>
%F A351351 a(n)/A351352(n) <= A351349(n)/A351350(n).
%e A351351 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 A351351 .
%e A351351   n = 8 (r^2 = 1/2 = a(8)/A351352(8)):
%e A351351   .  3  1
%e A351351   4  *  8
%e A351351   5  7  .
%e A351351 .
%e A351351   n = 9 (r^2 = 1 = a(9)/A351352(9)):
%e A351351   .  3  2  .  .
%e A351351   4  .  .  1  .
%e A351351   5  .  *  0  9
%e A351351   .  6  7  8  .
%e A351351 .
%e A351351   n = 10 (r^2 = 2 = a(10)/A351352(10)):
%e A351351   .  .  3  2  .
%e A351351   .  4  .  .  1
%e A351351   5  .  *  . 10
%e A351351   6  .  .  9  .
%e A351351   .  7  8  .  .
%e A351351 .
%e A351351   n = 12 (r^2 = 4 = a(12)/A351352(12)):
%e A351351   .  4  3  2  .
%e A351351   5  .  .  .  1
%e A351351   6  .  *  . 12
%e A351351   7  .  .  . 11
%e A351351   .  8  9 10  .
%e A351351 .
%e A351351   n = 13 (r^2 = 9 = a(13)/A351352(13)):
%e A351351   .  .  .  4  .  3  .  .  .  .
%e A351351   .  5  .  .  .  .  .  2  .  .
%e A351351   6  .  .  .  .  .  .  .  1  .
%e A351351   7  .  .  .  *  .  .  .  0 13
%e A351351   8  .  .  .  .  .  .  .  .  .
%e A351351   .  9  .  .  .  .  . 12  .  .
%e A351351   .  .  . 10  . 11  .  .  .  .
%Y A351351 Cf. A351042, A351349, A351350, A351352 (denominators).
%K A351351 nonn,frac,more
%O A351351 8,3
%A A351351 _Pontus von Brömssen_, Feb 09 2022