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 1-7 of 7 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).

A351106 Triangle read by rows: T(m,n) is the number of simple paths for a Racetrack car (using Moore neighborhood) with initial velocity zero, going from one corner to the diagonally opposite corner on an m X n grid, 1 <= n <= m.

Original entry on oeis.org

1, 1, 3, 1, 6, 23, 2, 17, 118, 1470, 3, 47, 762, 23878, 914525, 5, 133, 5724, 420894, 40285572
Offset: 1

Views

Author

Pontus von Brömssen, Jan 31 2022

Keywords

Examples

			Triangle begins:
  m\n| 1   2    3      4        5  6
  ---+------------------------------
  1  | 1
  2  | 1   3
  3  | 1   6   23
  4  | 2  17  118   1470
  5  | 3  47  762  23878   914525
  6  | 5 133 5724 420894 40285572  ?

Links

Crossrefs

Cf. A291896 (column n=1), A329118, A351041, A351107 (main diagonal), A351108, A351110.

A351107 Number of simple paths for a Racetrack car (using Moore neighborhood) with initial velocity zero, going from one corner to the diagonally opposite corner on an n X n grid.

Original entry on oeis.org

1, 3, 23, 1470, 914525
Offset: 1

Views

Author

Pontus von Brömssen, Feb 01 2022

Keywords

Examples

			For n = 3 the following paths exist (up to reflection in the diagonal). The numbers give the positions of the car after successive steps.
  ..2  ..3  ..3  ..3  ..4  ..4  .34  .56  456  548  678  678
  .1.  ..2  .2.  .12  ..3  .23  .2.  .43  32.  673  543  512
  0..  01.  01.  0..  012  01.  01.  012  01.  012  012  043
Of these, only the first path is symmetric with respect to the diagonal, so the other 11 give rise to 2 paths each. In total, there are a(3) = 1 + 2*11 = 23 possible paths.

Links

Crossrefs

Main diagonal of A351106.
Cf. A140518, A351041, A351109, A351111.

A351110 Triangle read by rows: T(m,n) is the number of paths for a Racetrack car (using Moore neighborhood) with initial velocity zero, going from one corner to the diagonally opposite corner on an m X n grid, such that all positions are visited exactly once, 1 <= n <= m.

Original entry on oeis.org

1, 1, 0, 1, 1, 6, 1, 0, 15, 2, 1, 1, 70, 289, 9436, 1, 0, 294, 191, 128020
Offset: 1

Views

Author

Pontus von Brömssen, Feb 01 2022

Keywords

Comments

For a Racetrack car using von Neumann neighborhood (see A351042), there are no such paths if 2 <= n <= m, because the car will never be able to leave a corner of the grid (except the corner where it starts).

Examples

			Triangle begins:
  m\n| 1  2   3   4      5  6
  ---+-----------------------
  1  | 1
  2  | 1  0
  3  | 1  1   6
  4  | 1  0  15   2
  5  | 1  1  70 289   9436
  6  | 1  0 294 191 128020  ?

Links

Crossrefs

Cf. A000012 (column n=1), A000035 (column n=2), A272445, A351041, A351042, A351106, A351111 (main diagonal).

A351111 Number of paths for a Racetrack car (using Moore neighborhood) with initial velocity zero, going from one corner to the diagonally opposite corner on an n X n grid, such that all positions are visited exactly once.

Original entry on oeis.org

1, 0, 6, 2, 9436
Offset: 1

Views

Author

Pontus von Brömssen, Feb 01 2022

Keywords

Examples

			For n = 4 the following path and its reflection in the diagonal are the only solutions, so a(4) = 2.
   _   _
  | | |_
  |_ \  |
   _| |_|

Links

Crossrefs

Main diagonal of A351110.
Cf. A272445, A351041, A351107.
Showing 1-7 of 7 results.

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