A308590 -id:A308590 - cp's OEIS frontend

A326352 Total number of black cells after n iterations of Langton's ant with two ants on the grid placed side-by-side with one empty square between them and initially looking in the same direction.

0, 2, 4, 6, 8, 6, 6, 6, 6, 8, 6, 8, 10, 12, 14, 12, 12, 12, 12, 14, 12, 12, 12, 12, 14, 12, 14, 16, 18, 20, 18, 18, 16, 16, 18, 16, 18, 20, 22, 24, 22, 22, 22, 22, 24, 22, 22, 22, 22, 24, 22, 24, 26, 28, 30, 28, 28, 26, 26, 28, 28, 30, 30, 32, 34, 34, 36, 34

Offset: 0

Felix Fröhlich, Jun 30 2019

nonn

The two ants meet seven times; in that case, the color of the current square is flipped only once. Eventually, both ants build a recurrent highway pattern. - Rémy Sigrist, Jul 28 2019

			See illustrations in Fröhlich, 2019.

Rémy Sigrist, Table of n, a(n) for n = 0..20000
Felix Fröhlich, Illustration of iterations 0-50, 2019.
Rémy Sigrist, Representation of the grid after 20000 iterations
Rémy Sigrist, PARI program for A326352
Wikipedia, Langton's ant

Cf. A255938, A269757, A308590, A325953, A325954, A325955, A326167, A326353.

PARI
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See Links section.
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a(n + 104) = a(n) + 24 for any n >= 14373. - Rémy Sigrist, Jul 28 2019

More terms from Rémy Sigrist, Jul 28 2019

A308937 Langton's ant on a chair tiling: number of black cells after n moves of the ant.

Original entry on oeis.org

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

Offset: 0

Felix Fröhlich, Jul 01 2019

The ant begins on the inner corner of a subtile.
On a white tile, turn 90 degrees right, flip the color of the tile, then move forward until reaching a new tile, moving as far as possible within the tile.
On a black tile, turn 90 degrees left, then continue as above.
The chair tiling used for this automaton is, like all aperiodic hierarchical tilings, not unique (see for example Goodman-Strauss, p. 490). See "Remarks, 2019" in links for clarification which tiling the ant lives on.

			See illustrations in Fröhlich, 2019.

Jinyuan Wang, Table of n, a(n) for n = 0..1000
Felix Fröhlich, Illustration of iterations 0-50 of the ant, 2019.
Felix Fröhlich, Remarks specifying the tiling used for generating the sequence, 2019.
Chaim Goodman-Strauss, Aperiodic Hierarchical Tilings, in: J. F. Sadoc and N. Rivier, Foams and Emulsions, NATO Science Series, Series E, Vol. 354, Springer, pp 481-496, DOI:10.1007/978-94-015-9157-7_28.
Tilings Encyclopedia, Chair
Wikipedia, Langton's ant
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A255938, A269757, A308590, A325953, A325954, A325955.

a(n) = a(n-42) for n >= 178. - Jinyuan Wang, Jul 13 2025

More terms from Jinyuan Wang, Jul 13 2025

A308973 Langton's ant on a truncated square tiling: number of black cells after n moves of the ant when starting on an octagon and looking towards an edge where the tile meets another octagon.

Original entry on oeis.org

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

Offset: 0

Felix Fröhlich, Jul 04 2019

nonn

First differs from A269757 at n = 19.
On a white square, turn 90 degrees right, flip the color of the tile, then move forward one unit.
On a white octagon, turn 45 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 octagon, turn 45 degrees left, flip the color of the tile, then move forward one unit.
As in the original variant, order emerges after a transition phase and the ant starts building a recurrent "highway" pattern of 12 steps that repeats indefinitely. - Rémy Sigrist, Jul 21 2019

			See illustrations in Fröhlich, 2019.

Rémy Sigrist, Table of n, a(n) for n = 0..500
Felix Fröhlich, Illustration of iterations 0-50 of the ant, 2019.
Rémy Sigrist, Configuration after 100 steps
Rémy Sigrist, PARI program for A308973
Wikipedia, Langton's ant
Wikipedia, Truncated square tiling

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

PARI
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See Links section.
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a(n + 12) = a(n) + 6 for any n >= 34. - Rémy Sigrist, Jul 21 2019

More terms from Rémy Sigrist, Jul 21 2019

A309064 Langton's ant on a snub square tiling: number of black cells after n moves of the ant when starting on a square.

Original entry on oeis.org

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

Offset: 0

Felix Fröhlich, Jul 10 2019

nonn

First differs from A276073 at n = 16.
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.

Lars Blomberg, Table of n, a(n) for n = 0..10000
Lars Blomberg, The state for n=104000, when 872 cells are set
Lars Blomberg, Animation illustrating n=1-3000
Lars Blomberg, Animation illustrating the transition from "chaos" to "avenue", n=96300-99608
Felix Fröhlich, Illustration of iterations 0-50 of the ant, 2019.
Wikipedia, Langton's ant
Wikipedia, Snub square tiling

Cf. A255938, A269757, A276073, A308590, A308937, A308973, A326167, A326352.

a(n+1025) = a(n) + 25 for n > 96420. Lars Blomberg, Aug 15 2019

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.

Original entry on oeis.org

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

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

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

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

cp's OEIS Frontend

A326352 Total number of black cells after n iterations of Langton's ant with two ants on the grid placed side-by-side with one empty square between them and initially looking in the same direction.

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A308937 Langton's ant on a chair tiling: number of black cells after n moves of the ant.

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A308973 Langton's ant on a truncated square tiling: number of black cells after n moves of the ant when starting on an octagon and looking towards an edge where the tile meets another octagon.

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A309064 Langton's ant on a snub square tiling: number of black cells after n moves of the ant when starting on a square.

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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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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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