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

8,3

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.

Links

Table of n, a(n) for n=8..32.

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

Wikipedia, Racetrack

Formula

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

Example

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  .  .  .  .

Crossrefs

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

Sequence in context: A322765 A281605 A199499 * A160126 A257515 A105152

Adjacent sequences: A351348 A351349 A351350 * A351352 A351353 A351354

Keyword

nonn,frac,more

Author

Pontus von Brömssen, Feb 09 2022

Status

approved