A351349 Numerator of the square of the radius of the largest circle, centered at the origin, around which a Racetrack car (using Moore neighborhood) can run a full lap in n steps.

1, 1, 1, 4, 4, 81, 9, 16, 16, 576, 36, 36, 64, 81, 1250, 100, 144, 144, 8100, 225

Offset

6,4

Comments

The car starts and finishes on the positive x-axis, as in A351041.

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=6..25.

Pontus von Brömssen, Some optimal Racetrack trajectories for A351349/A351350.

Wikipedia, Racetrack

Formula

a(n)/A351350(n) >= A351351(n)/A351352(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 = 6 (r^2 = 1/2 = a(6)/A351350(6)):
  .  1  .
  3  *  6
  4  5  .
.
  n = 7 (r^2 = 1 = a(7)/A351350(7)):
  .  2  .  1  .
  3  .  *  .  7
  .  5  .  6  .
.
  n = 9 (r^2 = 4 = a(9)/A351350(9)):
  .  3  .  2  .
  4  .  .  .  1
  .  .  *  .  9
  5  .  .  .  8
  .  6  .  7  .
.
  n = 11 (r^2 = 81/10 = a(11)/A351350(11)):
  .  4  .  3  .  .  .  .  .  .
  5  .  .  .  .  .  2  .  .  .
  .  .  .  .  .  .  .  .  1  .
  6  .  .  *  .  .  .  . 11  0
  .  .  .  .  .  .  .  .  .  .
  7  .  .  .  .  . 10  .  .  .
  .  8  .  9  .  .  .  .  .  .
.
  n = 12 (r^2 = 9 = a(12)/A351350(12)):
  .  .  .  4  .  3  .  .  .
  .  5  .  .  .  .  .  2  .
  .  .  .  .  .  .  .  .  1
  6  .  .  .  *  .  .  . 12
  7  .  .  .  .  .  .  .  .
  .  8  .  .  .  .  . 11  .
  .  .  .  9  . 10  .  .  .

Crossrefs

Cf. A351041, A351350 (denominators), A351351, A351352.

Sequence in context: A107053 A327303 A337302 * A222271 A068376 A131592

Adjacent sequences: A351346 A351347 A351348 * A351350 A351351 A351352

Keyword

nonn,frac,more

Author

Pontus von Brömssen, Feb 09 2022

Status

approved