A325953 Langton's ant on a three-dimensional grid: number of black cells on the grid after n moves of the ant.

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

Offset: 0

Author

Felix Fröhlich, May 28 2019

Keywords

• nonn

Comments

The ant starts on a completely white grid.

Order of operations: turn 90 degrees, change cell color, move forward one unit.

The rules governing the movement of the ant and the change of cell colors are shown in the following table:

Cell color at | Direction of | Color to which the

start of iteration | 90-degree turn | cell is changed

-------------------+----------------+-------------------

white | right | black

black | left | blue

blue | up | yellow

yellow | down | white

The first differences of this sequence are ultimately periodic with period 28 beginning at generation 93475, see A308563. - Charlie Neder, Jun 10 2019

Examples

			See illustration in links.
		

Links

• Jinyuan Wang, Table of n, a(n) for n = 0..10000
• Felix Fröhlich, Illustration of iterations 0-50 of the ant
• Charlie Neder, Python program for running this automaton
• Wikipedia, Langton's ant

Crossrefs

Cf. A255938, A308563, A325954, A325955.

Formula

a(n) = a(n-28) + 4 for n >= 93504. - Jinyuan Wang, Jul 13 2025

Extensions

a(51)-a(68) from Charlie Neder, Jun 06 2019