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.

A351041 Minimal number of steps for a Racetrack car (using Moore neighborhood) to go around a circle of radius n.

Original entry on oeis.org

7, 9, 12, 13, 15, 16, 18, 18, 19, 21, 22, 22, 24, 24, 25, 26, 27, 27, 28, 28, 30, 31, 31, 31, 32, 33, 33, 34, 35, 35, 36, 36, 37, 37, 37
Offset: 1

Views

Author

Pontus von Brömssen, Jan 29 2022

Keywords

Comments

The car moves according to the rules of the game of Racetrack, i.e., if P, Q, and R are three successive positions of the car, both coordinates of the second difference (acceleration vector) P - 2Q + R must be 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) = 7:
  .  2  .  1  .  .
  3  .  *  .  0  7
  .  5  .  6  .  .
  (The car stands still on the fourth step.)
.
  a(2) = 9:
  .  3  .  2  .  .
  4  .  .  .  1  .
  .  .  *  .  0  9
  5  .  .  .  8  .
  .  6  .  7  .  .
.
  a(3) = 12:
  .  .  .  4  3  .  .  .  .
  .  5  .  .  .  .  2  .  .
  .  .  .  .  .  .  .  .  .
  6  .  .  .  .  .  .  1  .
  7  .  .  .  *  .  .  0 12
  .  .  .  .  .  .  .  .  .
  .  8  .  .  .  .  . 11  .
  .  .  .  9  . 10  .  .  .

Links

Crossrefs

Cf. A000217, A002024, A027434, A351042, A351349, A351350.

Formula

a(n) = min {k >= 6; A351349(k)/A351350(k) >= n^2}.
a(n) <= A351042(n).
a(n) >= A027434(n) + A027434(2*n) + A002024(n). This can be seen by looking at the y-coordinate only: First, the car must go up to at least y = n and reduce the speed in the y-direction to zero in order to turn downwards; this requires at least A027434(n) steps. Then down to y = -n or below with speed reduced to zero; this requires at least A027434(2*n) steps. Finally, up to at least y = 0 (not necessarily reducing the speed); this requires at least A002024(n) steps.
It appears that a(n) = A027434(n) + A027434(2*n) + A002024(n) + 1 if n is a triangular number (A000217), otherwise a(n) = A027434(n) + A027434(2*n) + A002024(n).

A351352 Denominator 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

2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 2, 2, 5, 13, 1, 13, 1, 25, 1, 1, 5, 1, 1
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

			See A351351 for examples.

Links

Crossrefs

Cf. A351042, A351350, A351351 (numerators).

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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