TOPICS
Search

Rule 190

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
ElementaryCARule190

Rule 190 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 190=10111110_2. This rule is illustrated above together with the evolution of a single black cell it produces after 15 steps (Wolfram 2002, p. 55).

The mirror image, complement, and mirror complement are rules 246, 130, and 144, respectively.

Starting with a single black cell, successive generations n=0, 1, ... are given by interpreting the numbers 1, 7, 29, 119, 477, 1911, 7645, 30583, ... (OEIS A037576) in binary, namely 1, 111, 11101, 1110111, 111011101, ... (OEIS A118111). The nth term is given by the first n terms of the quaternary sequence 131313..., or, more explicitly, by

a(n)={7/(15)(4^(n+1)-1) for n odd; 1/(15)(7·4^(n+1)-13) for n even
(1)
=1/(15)[-10-3(-1)^n+7·4^(n+1)]
(2)

(E. W. Weisstein, Apr. 13, 2006). Rule 190 is therefore computationally reducible for an initial configuration consisting of a single black cell. It has generating function

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

See also

Elementary Cellular Automaton, Rule 30, Rule 50, Rule 54, Rule 60, Rule 62, Rule 90, Rule 94, Rule 102, Rule 110, Rule 126, Rule 150, Rule 158, Rule 188, Rule 220, Rule 222

Related Wolfram sites

http://atlas.wolfram.com/01/01/190/

Explore with Wolfram|Alpha

References

Sloane, N. J. A. Sequences A037576 and A118111 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 190

Cite this as:

Weisstein, Eric W. "Rule 190." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Rule190.html

Subject classifications