A284119 - OEIS

%I #48 Jul 25 2023 08:48:50

%S 4,15,10,25,411,47,2128,853,372,2805,366,2603,703,37912,612,127,998,

%T 2401,1200,623,5280,1778,1462,4346269,4129,3241,7018,3885,14632,7019,

%U 4564,4277,147688,1857,11120,81141,20204,3847,116014,7635,6488,5665,6142,73515,5826,6062,3781,7865,28630

%N Preperiod (or threshold) of orbit of Post's {00, 1101} tag system applied to the word (100)^n, or -1 if this word has an unbounded trajectory.

%C Post's tag system maps a word w over {0,1} to w', where if w begins with 0, w' is obtained by appending 00 to w and deleting the first three letters, or if w begins with 1, w' is obtained by appending 1101 to w and deleting the first three letters.

%C The empty word is included in the count.

%C The orbit of the word (100)^n for n=110 dies after 43913328040672 iterations. The longest word in the orbit is 31299218, which appeared at iteration 14392308412264. See also A291792. - _Lars Blomberg_, Oct 04, 2017

%H Lars Blomberg, <a href="/A284119/b284119.txt">Table of n, a(n) for n = 1..6075</a>

%H Peter R. J. Asveld, <a href="http://doc.utwente.nl/66184/1/1988m20.pdf">On a Post's System of Tag</a>. Bulletin of the EATCS 36 (1988), 96-102.

%H Lars Blomberg, <a href="/A284119/a284119.png">Graph of the terms</a>

%H Lars Blomberg, <a href="/A284119/a284119_1.png">Graph of preperiod of a(110)</a>

%H Lars Blomberg, <a href="/A284119/a284119_2.png">Graph of preperiod of a(4974)</a>

%H N. J. A. Sloane, Three (No, 8) Lovely Problems from the OEIS, Experimental Mathematics Seminar, Rutgers University, Oct 05 2017, <a href="https://vimeo.com/237029685">Part I</a>, <a href="https://vimeo.com/237030304">Part 2</a>, <a href="https://oeis.org/A290447/a290447_slides.pdf">Slides.</a> (Mentions this sequence)

%F From _Lars Blomberg_, Apr 20 2018: (Start)

%F Using Excel, trendlines were created for the preperiod of the Post Tag and Watanabe Tag systems as follows:

%F A284119: y = 8.6528*x^2.0831, R^2 = 0.478.

%F A292090: y = 8.5595*x^2.1033, R^2 = 0.472.

%F Although the error value is rather large, the curves are quite similar. (End)

%e For n = 2 the orbit of (100)^2 = 100100 consists of a preperiod of length 15, followed by a periodic portion of length 6:

%e Preperiod:

%e 100100,

%e 1001101,

%e 11011101,

%e 111011101,

%e 0111011101,

%e 101110100,

%e 1101001101,

%e 10011011101,

%e 110111011101,

%e 1110111011101,

%e 01110111011101,

%e 1011101110100,

%e 11011101001101,

%e 111010011011101,

%e 0100110111011101,

%e followed by the period:

%e (011011101110100,

%e 01110111010000,

%e 1011101000000,

%e 11010000001101,

%e 100000011011101,

%e 0000110111011101)

%e (repeat)

%Y Cf. A284116, A284121, A292090.

%K nonn

%O 1,1

%A _Jeffrey Shallit_, Mar 20 2017

%E Edited by _N. J. A. Sloane_, Jul 29 2017. Added "escape clause" to the definition, Apr 19 2018.