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Rule 50


ElementaryCARule50

Rule 50 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 50=001100100_2. This rule is illustrated above together with the evolution of a single black cell it produces after 15 steps (Wolfram 2002, p. 55).

ElementaryCARule50Others

Note that for initial conditions of a single black cell, rule 50 is equivalent to rules 58, 114, 122, 178, 186, 242, and 250, which are precisely those rules with binary representation xx11x010_2. Variants obtained by complementing and mirror reversing and complementing are rules 160, 161, 162, 163, 176, 177, 178, and 179.

Starting with a single black cell, successive generations n=0, 1, ... are given by interpreting the numbers 1, 5, 21, 85, 341, 1365, 5461, 21845, 87381, 349525, ... (OEIS A002450) in binary, namely 1, 101, 10101, ... (OEIS A071028). The nth term is given by

a(n)=sum_(k=0)^(n)2^(2k)
(1)
=1/3(4^(n+1)-1),
(2)

so computation of the nth generation is computationally reducible for an initial configuration consisting of a single black cell. a(n) has generating function

 1/((x-1)(4x-1)).
(3)

See also

Additive Cellular Automaton, Elementary Cellular Automaton, Rule 30, Rule 54, Rule 60, Rule 62, Rule 90, Rule 94, Rule 102, Rule 110, Rule 126, Rule 150, Rule 158, Rule 182, Rule 188, Rule 190, Rule 220, Rule 222, Rule 250

Related Wolfram sites

http://atlas.wolfram.com/01/01/50/

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References

Sloane, N. J. A. Sequences A002450/M3914 and A071028 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.

Referenced on Wolfram|Alpha

Rule 50

Cite this as:

Weisstein, Eric W. "Rule 50." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Rule50.html

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