Rule 30 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).
250 iterations of rule 30 are illustrated above.
Starting with a single black cell on an otherwise white infinite row, successive generations 
,
1, ... are given by interpreting the numbers 1, 7, 25, 111, 401, 1783, 6409, 28479,
102849, ... (OEIS A110240) in binary,
namely 1, 111, 11001, 1101111, 110010001, ... (OEIS A070950).
Rule 30 is the mirror image, complement, and mirror complement of rules 86, 135, and 149, respectively.
Rule 30 is of special interest because it is chaotic (Wolfram 2002, p. 871), with central column given by 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, ... (OEIS A051023). Rule 30 was used for integer random number generation in versions of Mathematica prior to 6.0 (Wolfram 2002, p. 317). It remains available as the "Rule30CA" method in the Wolfram Language, whose default generator is "ExtendedCA" (Wolfram Research). Interpreting the central column as binary numbers and taking successive bits gives the sequence of numbers 1, 3, 6, 13, 27, 55, 110, 220, 441, 883, 1766, ... (OEIS A092539). The members of this sequence that are prime are 3, 13, 883, 237051898781, ... (OEIS A092540).
Jen (1990) proved that with the initial state of a single black cell, the sequence of ordered pairs of colors attained in any two adjacent
cells is not ultimately periodic (Gray 2003). The numbers of black cells 
in consecutive generations 
, 1, ... are 1, 3, 3, 6, 4, 9, 5, 12, 7, ... (OEIS A070952),
which is very closely fit by the line 
.
Wolfram (2019) announced the Rule 30 prize problems, offering $10,000 each for questions concerning nonperiodicity, equal limiting frequencies, and the computation time required to determine the central column.
The maximum lengths of white runs between the outermost black cells at generations 0, 1, 2, ... are 0, 0, 2, 1, 3, 1, 4, 2, 5, 3, 4, 4, 3, 2, ... (OEIS A100053). The high-water marks are 0, 2, 3, 4, 5, 6, 8, 9, 11, 14, 15, 23, ... (OEIS A100054), which occur at positions 0, 2, 4, 6, 8, 16, 32, 43, 46, 64, 128, 256, 512, ... (OEIS A100055; Weisstein, Oct. 31, 2004) and look suspiciously like powers of 2 with a few additional values thrown in near the beginning.
An independent observation by E. Rowland (May 13, 2004) is that the sequence of lengths of the rightmost contiguous black runs is 1, 3, 1, 4, 1, 3, 1, 6, 1, 3,
1, 4, 1, 3, 1, 7, 1, 3, 1, 4, ... (OEIS A094603),
with high-water marks of 1, 3, 4, 6, 7, 9, 15, 16, 24, 25, 27, ... (OEIS A094604)
at generations 0, 1, 3, 7, 15, ... (OEIS A000225;
i.e., 
; Rowland 2006).
