A255938 -id:A255938 - cp's OEIS frontend

A309166 Langton's ant on a truncated hexagonal tiling: number of black cells after n moves of the ant when starting on a dodecagon and looking towards an edge where the dodecagon meets a triangle.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 13, 12, 11, 12, 13, 14, 15, 16, 15, 14, 15, 16, 17, 18, 19, 20, 19, 18, 17, 16, 17, 18, 19, 20, 21, 22, 23, 22, 23, 24, 25, 26, 27, 26, 27, 28, 29, 30, 31, 30, 29, 30, 31, 32, 33, 34, 33, 32, 33, 32

Offset: 0

Felix Fröhlich, Jul 15 2019

nonn

On a white dodecagon, turn 30 degrees right, flip the color of the tile, then move forward one unit.
On a black dodecagon, turn 30 degrees left, flip the color of the tile, then move forward one unit.
On a white triangle, turn 60 degrees right, flip the color of the tile, then move forward one unit.
On a black triangle, turn 60 degrees left, flip the color of the tile, then move forward one unit.

			See illustrations in Fröhlich, 2019.

Sean A. Irvine, Table of n, a(n) for n = 0..10000
Lars Blomberg, The state for n=2200, when 342 cells are set
Lars Blomberg, Animation illustrating n=1-2200
Felix Fröhlich, Illustration of iterations 0-50 of the ant, 2019.
Sean A. Irvine, Java program (github)
Wikipedia, Langton's ant
Wikipedia, Truncated hexagonal tiling

Cf. A255938, A269757, A308590, A308937, A308973, A326167, A326352, A309064.

a(n+15) = a(n) + 9 for n > 2034. - Lars Blomberg, Aug 13 2019

More terms from Sean A. Irvine, Jul 22 2019

A274370 Let the starting square of Langton's ant have coordinates (0, 0), with the ant looking in negative x-direction. a(n) is the y-coordinate of the ant after n moves.

Original entry on oeis.org

0, 1, 1, 0, 0, -1, -1, 0, 0, -1, -1, -2, -2, -1, -1, 0, 0, -1, -1, -2, -2, -3, -3, -2, -2, -1, -1, -2, -2, -1, -1, -2, -2, -1, -1, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 1, 1, 0, 0, 1, 1, 2, 2, 3, 3, 2, 2, 1, 1, 2, 2, 1, 1, 2

Offset: 0

Felix Fröhlich, Jun 19 2016

Rémy Sigrist, Table of n, a(n) for n = 0..15000
Felix Fröhlich, Coordinates of Langton's ant.
Wikipedia, Langton's ant.

Cf. A274369 (x-coordinate).

Cf. A102358, A102369, A204810, A255938, A261990, A269757.

a(n+104) = a(n) - 2 for n > 9975. - Andrey Zabolotskiy, Jul 05 2016

A309241 Langton's ant on a rhombitrihexagonal tiling: number of black cells after n moves of the ant when starting on a hexagon.

Original entry on oeis.org

0, 1, 2, 3, 4, 3, 4, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 11, 12, 13, 14, 15, 16, 15, 14, 13, 12, 11, 10, 11, 10, 9, 8, 7, 8, 9, 10, 11, 12, 13, 12, 13, 14, 13, 12, 13, 14, 15, 16, 15, 14, 15, 14, 13, 14, 15, 14, 15, 16, 17, 18, 17, 18, 19, 20, 21, 22, 21, 20, 21

Offset: 0

Felix Fröhlich, Jul 17 2019

nonn

On a white hexagon, turn 60 degrees right, flip the color of the cell, then move forward one unit.
On a black hexagon, turn 60 degrees left, flip the color of the cell, then move forward one unit.
On a white square, turn 90 degrees right, flip the color of the cell, then move forward one unit.
On a black square, turn 90 degrees left, flip the color of the cell, then move forward one unit.
On a white triangle, turn 60 degrees right, flip the color of the cell, then move forward one unit.
On a black triangle, turn 60 degrees left, flip the color of the cell, then move forward one unit.

			See illustrations in Fröhlich, 2019.

Lars Blomberg, Table of n, a(n) for n = 0..1000
Lars Blomberg, The state for n=185, when 33 cells are set which is the maximum within the cycle
Lars Blomberg, Animation illustrating n=1-1000
Felix Fröhlich, Illustration of iterations 0-50, 2019.
Wikipedia, Langton's ant
Wikipedia, Rhombitrihexagonal tiling

Cf. A255938, A269757, A308590, A308937, A308973, A326167, A326352, A309064, A309166.

a(n+448) = a(n). - Lars Blomberg, Aug 16 2019

More terms from Lars Blomberg, Aug 16 2019

A309279 Langton's ant on a truncated trihexagonal tiling: number of black cells after n moves of the ant when starting on a dodecagon.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 5, 6, 7, 8, 9, 10, 9, 10, 11, 12, 13, 14, 13, 14, 15, 16, 17, 18, 17, 18, 19, 20, 21, 22, 21, 22, 21, 22, 23, 24, 25, 24, 25, 26, 25, 26, 27, 28, 29, 30, 31, 32, 31, 32, 33, 34, 35, 36, 35, 36, 37, 38, 39, 40, 39, 40, 41, 42, 43, 44, 43, 44, 45, 46, 47, 48, 47, 46

Offset: 0

Felix Fröhlich, Jul 20 2019

nonn

On a white dodecagon, turn 30 degrees right, flip the color of the tile, then move forward one unit.
On a black dodecagon, turn 30 degrees left, flip the color of the tile, then move forward one unit.
On a white hexagon, turn 60 degrees right, flip the color of the tile, then move forward one unit.
On a black hexagon, turn 60 degrees left, flip the color of the tile, then move forward one unit.
On a white square, turn 90 degrees right, flip the color of the tile, then move forward one unit.
On a black square, turn 90 degrees left, flip the color of the tile, then move forward one unit.

			See illustrations in Fröhlich, 2019.

Felix Fröhlich, Illustration of iterations 0-50 of the ant, 2019.
Sean A. Irvine, Java program (github)
Wikipedia, Langton's ant
Wikipedia, Truncated trihexagonal tiling

Cf. A255938, A269757, A308590, A308937, A308973, A326167, A326352, A309064, A309166, A309241.

More terms from Sean A. Irvine, Jul 22 2019

A309293 Langton's ant on a snub trihexagonal tiling: number of black cells after n moves of the ant when starting on a hexagon.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 4, 5, 6, 7, 8, 9, 10, 11, 12, 11, 10, 11, 12, 13, 14, 15, 14, 15, 16, 17, 18, 19, 20, 21, 20, 19, 18, 17, 16, 15, 14, 13, 14, 13, 12, 11, 10, 9, 10, 11, 12, 13, 14, 15, 16, 17, 16, 15, 16, 17, 18, 17, 16, 17, 18, 19, 18, 17, 16, 15, 14, 15

Offset: 0

Felix Fröhlich, Jul 21 2019

nonn

On a white tile, turn 60 degrees right, flip the color of the tile, then move forward one unit.
On a black tile, turn 60 degrees left, flip the color of the tile, then move forward one unit.
The sequence has a cycle of length of 28292, that is, a(28292)=0 with the ant in the starting hexagon pointing in the start direction, so another cycle will follow. The maximum term in the cycle is a(8148)=174. - Lars Blomberg, Aug 01 2019

			See illustrations in Fröhlich, 2019.

Lars Blomberg, Table of n, a(n) for n = 0..28292
Lars Blomberg, The state for n=8148, when 174 cells are set
Lars Blomberg, Video illustrating a full cycle
Felix Fröhlich, Illustration of iterations 0-50 of the ant, 2019.
Wikipedia, Langton's ant
Wikipedia, Snub trihexagonal tiling

Cf. A255938, A269757, A308590, A308937, A308973, A326167, A326352, A309064, A309166, A309241, A309279.

More terms from Lars Blomberg, Aug 01 2019

A282425 The maximum number of steps Langton's ant can make confined to an n X n grid.

Original entry on oeis.org

0, 5, 16, 45, 84, 163, 260

Offset: 1

Rok Cestnik, Feb 14 2017

a(8) >= 338, a(9) >= 397, a(10) >= 502.
From Rok Cestnik, Aug 25 2017: (Start)
We are looking for the combination of grid configuration, ant orientation and ant position that yields the maximal number of steps before the ant leaves the grid. We consider all possible grid configurations and ant positions, but since the ant may move forward and backwards in time (see third considered symmetry below) we deduce that the maximal solution will always have the ant start from the edge of the grid.
For the sake of solution presentation, we consider these rules: at a white cell turn left, at a black cell turn right (vice versa results in the same behavior, just mirrored). Some cells might not get visited in a solution; therefore they are unconstrained, and we color then gray. We also take into consideration some symmetries of the ant to avoid presenting several maximal solutions that are just transformations of a single solution. That said, it is not impossible that two fundamentally different configurations would both have the same maximal number of steps.
Considered symmetries of the ant:
  1. rotational symmetry, e.g., we consider that the configuration
    +-----+-----+-----+              +-----+-----+-----+
    |  |  |BBBBB|     |              |     |BBBBB|     |
    |  v  |BBBBB|     |              |     |BBBBB| <-- |
    |     |BBBBB|     |              |     |BBBBB|     |
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|     |     |      is      |BBBBB|     |BBBBB|
    |BBBBB|     |     |  equivalent  |BBBBB|     |BBBBB|
    |BBBBB|     |     |      to      |BBBBB|     |BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
    |     |BBBBB|BBBBB|              |BBBBB|     |     |
    |     |BBBBB|BBBBB|              |BBBBB|     |     |
    |     |BBBBB|BBBBB|              |BBBBB|     |     |
    +-----+-----+-----+              +-----+-----+-----+
.
  2. mirror symmetry combined with color inversion, e.g., we consider that the configuration
    +-----+-----+-----+              +-----+-----+-----+
    |     |BBBBB|     |              |BBBBB|     |BBBBB|
    |     |BBBBB| <-- |              |B-->B|     |BBBBB|
    |     |BBBBB|     |              |BBBBB|     |BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|     |BBBBB|      is      |     |BBBBB|     |
    |BBBBB|     |BBBBB|  equivalent  |     |BBBBB|     |
    |BBBBB|     |BBBBB|      to      |     |BBBBB|     |
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|     |     |              |BBBBB|BBBBB|     |
    |BBBBB|     |     |              |BBBBB|BBBBB|     |
    |BBBBB|     |     |              |BBBBB|BBBBB|     |
    +-----+-----+-----+              +-----+-----+-----+
.
  3. reversing the arrow of time combined with inverting the color of the cell on which the ant is located and turning the ant according to the color of its (now inverted) cell (with the chosen rules, if white turn left, if black turn right), e.g., the configuration
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|     |BBBBB|              |BBBBB|BBBBB|BBBBB|
    |B-->B|     |BBBBB|              |BBBBB|BBBBB|BBBBB|
    |BBBBB|     |BBBBB|              |BBBBB|BBBBB|BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
    |     |BBBBB|     |   will end   |BBBBB|BBBBB|BBBBB|
    |     |BBBBB|     |   in state   |BBBBB|BBBBB|BBBBB|
    |     |BBBBB|     |              |BBBBB|BBBBB|BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|BBBBB|     |              |     |     |BBBBB|
    |BBBBB|BBBBB|     |              | <-- |     |BBBBB|
    |BBBBB|BBBBB|     |              |     |     |BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
.
  and
.
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|BBBBB|BBBBB|              |     |     |BBBBB|
    |BBBBB|BBBBB|BBBBB|              |  ^  |     |BBBBB|
    |BBBBB|BBBBB|BBBBB|              |  |  |     |BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|BBBBB|BBBBB|   will end   |     |BBBBB|     |
    |BBBBB|BBBBB|BBBBB|   in state   |     |BBBBB|     |
    |BBBBB|BBBBB|BBBBB|              |     |BBBBB|     |
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|     |BBBBB|              |BBBBB|BBBBB|     |
    |BB^BB|     |BBBBB|              |BBBBB|BBBBB|     |
    |BB|BB|     |BBBBB|              |BBBBB|BBBBB|     |
    +-----+-----+-----+              +-----+-----+-----+
.
  hence
.
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|     |BBBBB|              |BBBBB|BBBBB|BBBBB|
    |B-->B|     |BBBBB|              |BBBBB|BBBBB|BBBBB|
    |BBBBB|     |BBBBB|              |BBBBB|BBBBB|BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
    |     |BBBBB|     |      is      |BBBBB|BBBBB|BBBBB|
    |     |BBBBB|     |  equivalent  |BBBBB|BBBBB|BBBBB|
    |     |BBBBB|     |      to      |BBBBB|BBBBB|BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
    |BBBBB|BBBBB|     |              |BBBBB|     |BBBBB|
    |BBBBB|BBBBB|     |              |BB^BB|     |BBBBB|
    |BBBBB|BBBBB|     |              |BB|BB|     |BBBBB|
    +-----+-----+-----+              +-----+-----+-----+
(End)
Due to the ant's complex nature, its trajectory is hard to predict; therefore, an exhaustive search through the possible grid configurations must be performed, making this sequence computationally demanding.

Rok Cestnik, Solution for n = 2
Rok Cestnik, Solution for n = 3
Rok Cestnik, Solution for n = 4
Rok Cestnik, Solution for n = 5
Rok Cestnik, Solution for n = 6
Rok Cestnik, Solution for n = 7
Rok Cestnik, C++ code for A282425
Rok Cestnik, Best found solution for n = 8
Rok Cestnik, Best found solution for n = 9
Rok Cestnik, Best found solution for n = 10
Wikipedia, Langton's ant

Cf. A007764, A099733, A255938.

a(7) from Rok Cestnik, Aug 12 2017

A309236 Langton's ant on a circular grid with 4-fold rotational symmetry: number of black cells on the grid after n moves of the ant.

Original entry on oeis.org

0, 1, 2, 3, 2, 3, 4, 5, 4, 3, 2, 3, 2, 3, 4, 3, 4, 5, 6, 5, 6, 7, 6, 7, 6, 5, 4, 5, 6, 7, 6, 7, 8, 9, 8, 7, 6, 7, 6, 5, 4, 3, 4, 5, 6, 5, 6, 5, 6, 7, 6, 7, 6, 7, 6, 5, 4, 5, 6, 7, 6, 7, 8, 9, 8, 7, 6, 7, 6, 7, 6, 7, 8, 7, 8, 7, 8, 9, 10, 9, 8, 7, 6, 7, 6, 5, 4

Offset: 0

Felix Fröhlich, Jul 17 2019

nonn

On a white circular segment, turn right to the next edge of the segment, flip the color of that segment, then move onto the segment adjacent to that edge.

On a black circular segment, turn left to the next edge of the segment, flip the color of that segment, then move onto the segment adjacent to that edge.

			See illustrations in Fröhlich, 2019.

Felix Fröhlich, Illustration of iterations 0-50, 2019.

Cf. A255938, A269757, A308590, A308937, A308973, A309064, A326167, A326352.

lista(nn) = my(c, d=1, x, y, u=1, v=List([])); print1(c); for(n=1, nn, if(x, if(x>#v, listput(v, [1, 1, 1, 1])); if(v[x][y]<0, d=d%4+1, d=(d+2)%4+1); c-=v[x][y]=-v[x][y]; if(d==3, x--; if(!x, d=(y+1)%4+1), x+=d%2; y=(y-d)%4+1), if(u<0, y=(d+2)%4+1, y=d%4+1); c-=u=-u; x=d=1); print1(", ", c)); \\ Jinyuan Wang, Jul 15 2025

More terms from Jinyuan Wang, Jul 15 2025

A309384 Number of black squares after n moves of a variant of Langton's ant with turns of 45 degrees.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 7, 8, 9, 10, 11, 12, 11, 10, 11, 12, 13, 14, 15, 14, 13, 14, 15, 16, 17, 18, 19, 20, 21, 20, 21, 22, 23, 24, 25, 24, 23, 24, 25, 26, 25, 24, 25, 26, 27, 28, 29, 28, 29, 30, 29, 30, 31, 30, 31, 32, 33, 34, 35, 34, 35, 36, 37, 36, 37

Offset: 0

Rémy Sigrist, Jul 27 2019

nonn

Initially, all squares are white, and the ant is at the origin and oriented towards the east. The ant can travel horizontally or vertically (in any of the four cardinal directions) and diagonally (in any of the four intercardinal directions).
At a white square, the ant turns 45 degrees left, flips the color of the square and moves forward.
At a black square, the ant turns 45 degrees right, flips the color of the square and moves forward.
As in the original variant, the ant eventually builds a recurrent highway pattern, in the present case of 196 steps.

			The first positions of the ant are:
  .  .  4  3  .  .  .  .
  .  5  .  .  2  .  .  .
  .  6  .  .  1  .  .  .
  .  .  7 0,8 .  . 11  .
  .  .  .  .  9 10  .  .

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Representation of the squares visited during the first 100000000 moves (the dark squares are the most visited)
Rémy Sigrist, Representation of the grid after 1021259426 steps
Rémy Sigrist, PARI program for A309384
Wikipedia, Langton's ant

Cf. A255938.

PARI
```
See Links section.
```

a(n + 196) = a(n) + 72 for n >= 1021254426.

A325631 Langton's ant on an elongated triangular tiling: number of black cells after n moves of the ant when starting on a square and initially looking towards one of the edges where that square meets one of the neighboring triangles.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 4, 5, 6, 7, 8, 9, 8, 9, 10, 11, 12, 13, 12, 13, 14, 15, 14, 13, 14, 15, 16, 15, 16, 17, 16, 15, 16, 17, 18, 17, 16, 15, 16, 17, 18, 19, 20, 19, 18, 19, 20, 19, 20, 21, 22, 23, 24, 23, 22, 23, 22, 23, 22, 21, 20, 19, 18, 19, 18, 17, 18, 19, 20

Offset: 0

Felix Fröhlich, Sep 07 2019

nonn

First differs from A276073 at n = 22.
On a white square, turn 90 degrees right, flip the color of the tile, then move forward one unit.
On a white triangle, turn 60 degrees right, flip the color of the tile, then move forward one unit.
On a black square, turn 90 degrees left, flip the color of the tile, then move forward one unit.
On a black triangle, turn 60 degrees left, flip the color of the tile, then move forward one unit.

			See illustrations in Fröhlich, 2019.

Jinyuan Wang, Table of n, a(n) for n = 0..2000
Felix Fröhlich, Illustration of iterations 0-50 of the ant, 2019.
Wikipedia, Elongated triangular tiling
Wikipedia, Langton's ant

Cf. A255938, A269757, A308590, A308937, A308973, A326167, A326352, A309064, A309166, A309241, A309279, A309293.

a(n) = a(n-51) + 11 for n >= 1159. - Jinyuan Wang, Jul 15 2025

More terms from Jinyuan Wang, Jul 15 2025

A275117 Direction where Langton's ant is looking after n moves: 1 if looking in starting direction, 2 if looking 90 degrees clockwise from starting direction, 3 if looking 90 degrees counterclockwise from starting direction, or 4 if looking in direction opposite to starting direction.

Original entry on oeis.org

1, 2, 4, 3, 1, 3, 1, 2, 4, 3, 4, 3, 1, 2, 4, 2, 1, 3, 4, 3, 4, 3, 1, 2, 4, 2, 4, 3, 1, 2, 1, 3, 4, 2, 4, 2, 4, 3, 1, 2, 1, 2, 4, 3, 1, 3, 4, 2, 1, 2, 1, 3, 1, 2, 1, 2, 4, 3, 1, 3, 4, 2, 1, 2, 1, 2, 4, 3, 1, 3, 1, 2, 4, 3, 4, 2, 1, 3, 1, 3, 1, 2, 4, 3, 4, 3, 1

Offset: 0

Felix Fröhlich, Jul 18 2016

nonn

Cf. A102358, A102369, A204810, A255938, A261990, A269757.

From Andrey Zabolotskiy, Oct 11 2016: (Start)
Let d(n) = (A255938(n) mod 4). Then:
a(n)=1 if d(n)=0,
a(n)=2 if d(n)=1,
a(n)=4 if d(n)=2,
a(n)=3 if d(n)=3.
(End)

More terms from Andrey Zabolotskiy, Oct 11 2016

cp's OEIS Frontend

A309166 Langton's ant on a truncated hexagonal tiling: number of black cells after n moves of the ant when starting on a dodecagon and looking towards an edge where the dodecagon meets a triangle.

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A274370 Let the starting square of Langton's ant have coordinates (0, 0), with the ant looking in negative x-direction. a(n) is the y-coordinate of the ant after n moves.

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A309241 Langton's ant on a rhombitrihexagonal tiling: number of black cells after n moves of the ant when starting on a hexagon.

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A309279 Langton's ant on a truncated trihexagonal tiling: number of black cells after n moves of the ant when starting on a dodecagon.

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A309293 Langton's ant on a snub trihexagonal tiling: number of black cells after n moves of the ant when starting on a hexagon.

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A282425 The maximum number of steps Langton's ant can make confined to an n X n grid.

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A309236 Langton's ant on a circular grid with 4-fold rotational symmetry: number of black cells on the grid after n moves of the ant.

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A309384 Number of black squares after n moves of a variant of Langton's ant with turns of 45 degrees.

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A325631 Langton's ant on an elongated triangular tiling: number of black cells after n moves of the ant when starting on a square and initially looking towards one of the edges where that square meets one of the neighboring triangles.

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