Rule 222

Rule 222 is one of the elementary cellular automaton rules introduced by Stephen Wolfram in 1983 (Wolfram 1983, 2002). It specifies the next color in a cell, depending on its color and its immediate neighbors. Its rule outcomes are encoded in the binary representation . This rule is illustrated above together with the evolution of a single black cell it produces after 15 steps (Wolfram 2002, p. 55).

Rule 222 is amphichiral, and its complement is rule 132.

Starting with a single black cell, successive generations , 1, ... are given by interpreting the numbers 1, 7, 31, 127, 511, 2047, 8191, ... (OEIS A083420) in binary, namely 1, 111, 11111, 1111111, 111111111, .... The th term is given by

which are Mersenne numbers, so rule 222 is computationally reducible for an initial configuration consisting of a single black cell.

REFERENCES:

Sloane, N. J. A. Sequence A083420 in "The On-Line Encyclopedia of Integer Sequences."

Wolfram, S. "Statistical Mechanics of Cellular Automata." Rev. Mod. Phys. 55, 601-644, 1983.

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. 55, 90, and 952, 2002.