cp's OEIS Frontend

A284119 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.

%I A284119 #50 Jun 27 2025 23:18:13
%S A284119 4,15,10,25,411,47,2128,853,372,2805,366,2603,703,37912,612,127,998,
%T A284119 2401,1200,623,5280,1778,1462,4346269,4129,3241,7018,3885,14632,7019,
%U A284119 4564,4277,147688,1857,11120,81141,20204,3847,116014,7635,6488,5665,6142,73515,5826,6062,3781,7865,28630
%N A284119 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 A284119 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 A284119 The empty word is included in the count.
%C A284119 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 A284119 Lars Blomberg, <a href="/A284119/b284119.txt">Table of n, a(n) for n = 1..6075</a>
%H A284119 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 A284119 Lars Blomberg, <a href="/A284119/a284119.png">Graph of the terms</a>
%H A284119 Lars Blomberg, <a href="/A284119/a284119_1.png">Graph of preperiod of a(110)</a>
%H A284119 Lars Blomberg, <a href="/A284119/a284119_2.png">Graph of preperiod of a(4974)</a>
%H A284119 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 A284119 From _Lars Blomberg_, Apr 20 2018: (Start)
%F A284119 Using Excel, trendlines were created for the preperiod of the Post Tag and Watanabe Tag systems as follows:
%F A284119 A284119: y = 8.6528*x^2.0831, R^2 = 0.478.
%F A284119 A292090: y = 8.5595*x^2.1033, R^2 = 0.472.
%F A284119 Although the error value is rather large, the curves are quite similar. (End)
%e A284119 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 A284119 Preperiod:
%e A284119 100100,
%e A284119 1001101,
%e A284119 11011101,
%e A284119 111011101,
%e A284119 0111011101,
%e A284119 101110100,
%e A284119 1101001101,
%e A284119 10011011101,
%e A284119 110111011101,
%e A284119 1110111011101,
%e A284119 01110111011101,
%e A284119 1011101110100,
%e A284119 11011101001101,
%e A284119 111010011011101,
%e A284119 0100110111011101,
%e A284119 followed by the period:
%e A284119 (011011101110100,
%e A284119 01110111010000,
%e A284119 1011101000000,
%e A284119 11010000001101,
%e A284119 100000011011101,
%e A284119 0000110111011101)
%e A284119 (repeat)
%Y A284119 Cf. A284116, A284121, A292090.
%K A284119 nonn
%O A284119 1,1
%A A284119 _Jeffrey Shallit_, Mar 20 2017
%E A284119 Edited by _N. J. A. Sloane_, Jul 29 2017. Added "escape clause" to the definition, Apr 19 2018.