A102358 -id:A102358 - cp's OEIS frontend

A102369 Intervals between successive arrivals of Langton's Ant at the origin.

Original entry on oeis.org

4, 4, 8, 36, 8, 36, 44, 44, 92, 92, 8, 8, 8, 8, 36, 8, 36, 184, 4, 2752, 732, 8, 2016, 596, 1284, 4, 4, 4, 8

Offset: 0

Robert H Barbour, Feb 22 2005, corrected Feb 27 2005

Ant Farm algorithm available from Robert H Barbour.

First differences of A102358. - Felix Fröhlich, Jul 26 2016

Christopher G. Langton et al., (1990) Artificial Life II. Addison-Wesley, Reading, MA., USA

P. Sarkar, A Brief History of Cellular Automata, ACM Computing Surveys. Vol. 32 No. 1 March (2000).

Cf. A102127, A102358.

A126978 a(n) = 104*n + 9977.

Original entry on oeis.org

9977, 10081, 10185, 10289, 10393, 10497, 10601, 10705, 10809, 10913, 11017, 11121, 11225, 11329, 11433, 11537, 11641, 11745, 11849, 11953, 12057, 12161, 12265, 12369, 12473, 12577, 12681, 12785, 12889, 12993, 13097, 13201, 13305, 13409, 13513, 13617, 13721, 13825

Offset: 0

Robert H Barbour, Mar 20 2007, Jun 12 2007

Langton's Ant Superhighway, the start point (9977th iteration, J. Propp) and the period length for the Superhighway (104).

B. D. Swan, Table of n, a(n) for n = 0..10000
C. Langton, Studying Artificial Life with Cellular Automata, Physica D: Nonlinear Phenomena, Vol. 22, 1986, pp. 120-149.
Ed Pegg Jr, 2D Turing Machines, 2004.
James Propp, Further Ant-ics, Mathematical Intelligencer, Vol. 16, 1994, pp. 37-42.
P. Sarkar, A Brief History of Cellular Automata, ACM Computing Surveys. Vol. 32, No. 1, Mar 01 2000, pp. 80-107.
S. Wolfram, 2D Turing Machines.
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A102127, A102358, A102369, A126979, A126980.

[104*n + 9977: n in [0..40]]; // Vincenzo Librandi, Sep 10 2015

104*Range[0,40]+9977 (* or *) LinearRecurrence[{2,-1},{9977,10081},40] (* Harvey P. Dale, Dec 16 2011 *)
CoefficientList[Series[(9977 - 9873 x)/(1 - x)^2, {x, 0, 40}], x] (* Vincenzo Librandi, Sep 10 2015 *)

a(n)=104*n+9977 \\ Charles R Greathouse IV, Oct 07 2015

a(0)=9977, a(1)=10081, a(n) = 2*a(n-1) - a(n-2). - Harvey P. Dale, Dec 16 2011
G.f.: (9977 - 9873*x)/(1-x)^2. - Vincenzo Librandi, Sep 10 2015
E.g.f.: exp(x)*(9977 + 104*x). - Elmo R. Oliveira, Dec 08 2024

A274369 Let the starting square of Langton's ant have coordinates (0, 0), with the ant looking in negative x-direction. a(n) is the x-coordinate of the ant after n moves.

Original entry on oeis.org

0, 0, 1, 1, 0, 0, -1, -1, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 2, 2, 1, 1, 2, 2, 3, 3, 2, 2, 1, 1, 2, 2, 3, 3, 4, 4, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 2, 2, 1, 1, 0, 0, -1, -1, 0, 0, -1, -1, 0, 0, -1, -1, -2, -2, -1, -1, -2, -2, -3, -3, -2, -2, -1, -1, -2, -2, -3

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. A274370 (y-coordinate).

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

# A274369: Langton's ant by Andrey Zabolotskiy, Jul 05 2016
def ant(n):
    steps = [(1, 0), (0, 1), (-1, 0), (0, -1)]
    black = set()
    x = y = 0
    position = [(x, y)]
    direction = 2
    for _ in range(n):
        if (x, y) in black:
            black.remove((x, y))
            direction += 1
        else:
            black.add((x, y))
            direction -= 1
        (dx, dy) = steps[direction%4]
        x += dx
        y += dy
        position.append((x, y))
    return position
print([p[0] for p in ant(100)])
# change p[0] to p[1] to get y-coordinates

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

A275302 Iterations at which Langton's Ant living on triangular tiling passes through the origin.

Original entry on oeis.org

0, 6, 24, 30, 72, 78, 96, 102, 108, 174, 180, 198, 212, 222, 252, 282, 292, 306, 324, 330, 408, 414, 420, 438, 444, 522, 544, 554, 576, 594, 648, 666, 672, 798, 804, 810, 852, 858, 920, 926, 972, 978, 984, 1018, 1024, 1154, 1160, 1178, 1184, 1190, 1208, 1214

Offset: 1

Oleg Nikulin, Jul 22 2016

nonn

Langton's Ant living on triangular tiling (or, equivalently, hexagonal grid) follows the rules similar to those of the ordinary Langton's ant. On a white cell, turn 60 degrees right, flip the color of the cell, then move forward one unit. On a black cell, turn 60 degrees left, flip the color of the cell, then move forward one unit.
On these iterations pattern becomes symmetric. Orientation of the ant on these iterations is always the same.
Empirically, a(n) ~ c*n^1.207.

Andrey Zabolotskiy, Table of n, a(n) for n = 1..25000 (calculated using _Oleg Nikulin_'s program)
Wikipedia, Turmite

Cf. A102358, A204810, A269757, A275303, A275304, A275305.

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

A308563 Langton's ant on a three-dimensional grid: iterations where the ant passes through the origin.

Original entry on oeis.org

0, 4, 8, 18, 130, 2206, 4326, 4650, 16344, 16814, 47942, 48000, 49928

Offset: 1

Felix Fröhlich and Charlie Neder, Jun 07 2019

For the rules applying to this ant, see A325953.

The sequence is finite, with 49928 being the last term. The ant never reaches the origin again after that, since it starts building a highway pattern at iteration 93475.

Charlie Neder, Python program for generating this sequence

Cf. A102358, A325953, A325954, A325955.

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

A102369 Intervals between successive arrivals of Langton's Ant at the origin.

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A126978 a(n) = 104*n + 9977.

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A274369 Let the starting square of Langton's ant have coordinates (0, 0), with the ant looking in negative x-direction. a(n) is the x-coordinate of the ant after n moves.

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A275302 Iterations at which Langton's Ant living on triangular tiling passes through the origin.

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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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A308563 Langton's ant on a three-dimensional grid: iterations where the ant passes through the origin.

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Comments

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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.

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