A102369 -id:A102369 - cp's OEIS frontend

A102358 Finite sequence of iterations at which Langton's Ant passes through the origin.

0, 4, 8, 16, 52, 60, 96, 140, 184, 276, 368, 376, 384, 392, 428, 436, 472, 656, 660, 3412, 4144, 4152, 6168, 6764, 8048, 8052, 8056, 8060, 8068

Offset: 1

Robert H Barbour, Feb 21 2005, corrected Feb 27 2005

Ant Farm algorithm available from bbarbour(AT)unitec.ac.nz.

Numbers n such that A274369(n) = A274370(n) = 0. - 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, A102369, A274369, A274370.

A102127 Interval between successive arrivals at the origin of an LQTL CA.

Original entry on oeis.org

4, 4, 4, 16, 4, 4, 4, 16, 4, 4, 4, 4, 4, 40, 4, 16, 4, 4, 4, 4, 12, 4, 4, 8, 4, 40, 4, 4, 4, 4, 4, 88, 4, 40, 4, 4, 4, 4, 4, 16, 4, 4, 4, 16, 4, 4, 4, 4, 4, 40, 4, 88, 4, 4, 4, 4, 4, 40, 4, 8, 4, 12, 4, 4, 4, 4, 4, 16, 4, 40, 4, 4, 4, 4, 4, 16, 4, 4, 4, 16, 4

Offset: 0

Robert H Barbour, Feb 14 2005

nonn

The "LQTL CA" is similar to the Langton's ant but has 4 states. The ant turns right from the cells with states 0, 1 and left from the cells with states 2, 3 and changes the state of the cell according to the rules 0 -> 1 -> 2 -> 3 -> 0. - Andrey Zabolotskiy, Jan 06 2023

Andrey Zabolotskiy, Table of n, a(n) for n = 0..999
Robert H. Barbour, Two-dimensional Four Color Cellular Automaton: Surface Explorations, Complex Systems, 17 (2007), 103-112.
Palash Sarkar, A Brief History of Cellular Automata, ACM Computing Surveys, Vol. 32, No. 1, March 2000, 80-107.

Cf. A094266, A094867, A099423, A274369, A102369.

First differences of A102110.

x = y = direction = 0
cells, a = {}, []
for n in range(1, 1000):
    c = cells.get((x, y), 0)
    cells[(x, y)] = (c + 1) % 4
    direction += [1, 1, -1, -1][c]
    (dx, dy) = [(1, 0), (0, 1), (-1, 0), (0, -1)][direction%4]
    x += dx;  y += dy
    if (x, y) == (0, 0):
        a.append(n)
print(a) # A102110
print([x-y for x, y in zip(a, [0]+a)]) # this sequence -  Andrey Zabolotskiy, Jan 06 2023

Terms a(26) and beyond from Andrey Zabolotskiy, Jan 06 2023

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

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

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

A102358 Finite sequence of iterations at which Langton's Ant passes through the origin.

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A102127 Interval between successive arrivals at the origin of an LQTL CA.

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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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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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Links

Crossrefs

Formula

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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Extensions