OFFSET
3,2
COMMENTS
From Roman Khrabrov, Aug 17 2024: (Start)
It appears that 2^A007814(n) * (2^A309786(n) - 1) divides a(n). For rule 90, it follows from Lemma 3.5 and Theorem 3.5 from Martin & Odlyzko & Wolfram's paper, and the definition of A309786. Rule 150 appears to have the same behavior (verified for n <= 1000).
The numbers for which a(n) differs from 2^A007814(n) * (2^A309786(n) - 1), are the powers of 2 and the numbers in the form 6*2^k, 13*2^k, 37*2^k, 61*2^k, 67*2^k, 95*2^k and so on (there is no corresponding OEIS sequence).
It seems that in 2D case (totalistic rule 34 on a toroidal grid) the formula 2^A007814(n) * (2^A309786(n) - 1) gives the correct maximum cycle lengths in all cases except powers of 2. Replacing A007814(n) with A091090(n) appears to always provide the correct maximum cycle lengths, even at powers of 2.
Conjecture: a(n) = n only if n belongs to A115770. The inverse does not hold true in general; the first exception is 445. (End)
REFERENCES
O. Martin, A. M. Odlyzko and S. Wolfram, Algebraic properties of cellular automata, Comm Math. Physics, 93 (1984), pp. 219-258, Reprinted in Theory and Applications of Cellular Automata, S Wolfram, Ed., World Scientific, 1986, pp. 51-90 and in Cellular Automata and Complexity: Collected Papers of Stephen Wolfram, Addison-Wesley, 1994, pp. 71-113 See Table 1.
LINKS
Roman Khrabrov, Table of n, a(n) for n = 3..1000
Shin-ichi Tadaki, Orbits in one-dimensional finite linear cellular automata, arXiv:cond-mat/9305012, 1993.
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Jul 03 2003
EXTENSIONS
More terms from Sean A. Irvine, Jun 10 2018
Name clarified by Roman Khrabrov, Aug 17 2024
STATUS
approved