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.

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

Author

Felix Fröhlich, Jun 19 2016

Keywords

• sign
• look

Links

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

Crossrefs

Cf. A274370 (y-coordinate).

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

Programs

Python

# 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

Formula

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