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.

1, 1, 2, 2, 4, 9, 9, 9, 16, 32, 32, 196, 81, 125, 392, 1225, 100, 1681, 160, 4489, 200, 225, 1369, 320, 400

Offset: 8

Author

Pontus von Brömssen, Feb 09 2022

Keywords

• nonn
• frac
• more

Comments

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

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.

Examples

			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.
.
  n = 8 (r^2 = 1/2 = a(8)/A351352(8)):
  .  3  1
  4  *  8
  5  7  .
.
  n = 9 (r^2 = 1 = a(9)/A351352(9)):
  .  3  2  .  .
  4  .  .  1  .
  5  .  *  0  9
  .  6  7  8  .
.
  n = 10 (r^2 = 2 = a(10)/A351352(10)):
  .  .  3  2  .
  .  4  .  .  1
  5  .  *  . 10
  6  .  .  9  .
  .  7  8  .  .
.
  n = 12 (r^2 = 4 = a(12)/A351352(12)):
  .  4  3  2  .
  5  .  .  .  1
  6  .  *  . 12
  7  .  .  . 11
  .  8  9 10  .
.
  n = 13 (r^2 = 9 = a(13)/A351352(13)):
  .  .  .  4  .  3  .  .  .  .
  .  5  .  .  .  .  .  2  .  .
  6  .  .  .  .  .  .  .  1  .
  7  .  .  .  *  .  .  .  0 13
  8  .  .  .  .  .  .  .  .  .
  .  9  .  .  .  .  . 12  .  .
  .  .  . 10  . 11  .  .  .  .
		

Links

• Pontus von Brömssen, Some optimal Racetrack trajectories for A351351/A351352.
• Wikipedia, Racetrack

Crossrefs

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

Formula

a(n)/A351352(n) <= A351349(n)/A351350(n).