%I #19 Aug 23 2024 16:11:34

%S 1,2,3,1,7,4,7,6,31,2,21,14,15,8,15,14,511,12,63,62,2047,4,1023,42,

%T 511,28,16383,30,31,16,31,30,4095,28,29127,1022,4095,24,1023,126,127,

%U 124,4095,4094,8388607,8,2097151,2046,255,84,67108863,1022,1048575,56,511,32766,536870911,60,17043521,62,63,32,63,62

%N Eventual period of a single cell in rule 150 cellular automaton in a cyclic universe of width n.

%C From _Roman Khrabrov_, Aug 17 2024: (Start)

%C 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).

%C 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).

%C 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.

%C Conjecture: a(n) = n only if n belongs to A115770. The inverse does not hold true in general; the first exception is 445. (End)

%D 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.

%H Roman Khrabrov, <a href="/A085588/b085588.txt">Table of n, a(n) for n = 3..1000</a>

%H Shin-ichi Tadaki, <a href="https://arxiv.org/abs/cond-mat/9305012">Orbits in one-dimensional finite linear cellular automata</a>, arXiv:cond-mat/9305012, 1993.

%Y Cf. A085587-A085595.

%K nonn

%O 3,2

%A _N. J. A. Sloane_, Jul 03 2003

%E More terms from _Sean A. Irvine_, Jun 10 2018

%E Name clarified by _Roman Khrabrov_, Aug 17 2024