cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A351042 Minimal number of steps for a Racetrack car (using von Neumann neighborhood) to go around a circle of radius n.

Original entry on oeis.org

9, 12, 13, 16, 17, 19, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 32, 32, 34, 34, 36, 36, 37
Offset: 1

Views

Author

Pontus von Brömssen, Jan 30 2022

Keywords

Comments

The car moves according to the rules of the game of Racetrack with von Neumann neighborhood, i.e., if P, Q, and R are three successive positions of the car, one coordinate of the second difference (acceleration vector) P - 2Q + R must be 0, and the other 1, 0, or -1. The car starts with zero velocity at a point (x,0) for some integer x >= n, and finishes when it passes, or lands on, the positive x-axis after a complete counterclockwise lap around the origin. The line segments between successive positions must be outside or on the circle with center in (0,0) and radius n.

Examples

			The following diagrams show examples of optimal trajectories for n = 1, 2, 3. The origin is marked with an asterisk.
.
  a(1) = 9:
  .  3  2  .  .
  4  .  .  1  .
  5  .  *  0  9
  .  6  7  8  .
.
  a(2) = 12:
  .  4  3  2  .  .
  5  .  .  .  1  .
  6  .  *  .  0 12
  7  .  .  . 11  .
  .  8  9 10  .  .
.
  a(3) = 13:
  .  .  .  4  .  3  .  .  .  .
  .  5  .  .  .  .  .  2  .  .
  6  .  .  .  .  .  .  .  1  .
  7  .  .  .  *  .  .  .  0 13
  8  .  .  .  .  .  .  .  .  .
  .  9  .  .  .  .  . 12  .  .
  .  .  . 10  . 11  .  .  .  .

Links

Crossrefs

Cf. A027434, A351041, A351043, A351351, A351352.

Formula

a(n) = min {k >= 8; A351351(k)/A351352(k) >= n^2}.
a(n) >= A351041(n).

A351350 Denominator 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.

Original entry on oeis.org

2, 1, 1, 1, 1, 10, 1, 1, 1, 17, 1, 1, 1, 1, 13, 1, 1, 1, 37, 1
Offset: 6

Views

Author

Pontus von Brömssen, Feb 09 2022

Keywords

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.

Examples

			See A351349 for examples.

Links

Crossrefs

Cf. A351041, A351349 (numerators), A351352.

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.

Original entry on oeis.org

1, 1, 1, 4, 4, 81, 9, 16, 16, 576, 36, 36, 64, 81, 1250, 100, 144, 144, 8100, 225
Offset: 6

Views

Author

Pontus von Brömssen, Feb 09 2022

Keywords

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.

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 = 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  .  .  .

Links

Crossrefs

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

Formula

a(n)/A351350(n) >= A351351(n)/A351352(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.

Original entry on oeis.org

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

Views

Author

Pontus von Brömssen, Feb 09 2022

Keywords

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

Crossrefs

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

Formula

a(n)/A351352(n) <= A351349(n)/A351350(n).
Showing 1-4 of 4 results.

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