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A292089 Numbers n such that Watanabe's 3-shift tag system {00/1011} started at the word (100)^n eventually dies (i.e., reaches the empty string).

%I A292089 #41 Mar 14 2025 04:25:53
%S A292089 5,9,11,16,20,22,23,25,37,38,43,47,61,64,66,68,71,82,87,95,100,115,
%T A292089 119,120,123,126,137,141,142,143,144,147,149,153,156,158,164,165,171,
%U A292089 178,179,183,188,195,196,201,202,203,205,206,212,214,216,218,223,232
%N A292089 Numbers n such that Watanabe's 3-shift tag system {00/1011} started at the word (100)^n eventually dies (i.e., reaches the empty string).
%C A292089 Watanabe's tag system {00/1011} 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 1011 to w and deleting the first three letters.
%C A292089 These are the numbers such that A292091(n)=0.
%C A292089 Oct 11, 2017: _Lars Blomberg_ has found that 872 is a member of this sequence. The word (100)^872 reaches the empty string after 72392976118788 iterations. The attached graph shows the lengths of the successive words in the trajectory. - _N. J. A. Sloane_, Oct 13 2017
%H A292089 Lars Blomberg, <a href="/A292089/b292089.txt">Table of n, a(n) for n = 1..1440</a>
%H A292089 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 A292089 Lars Blomberg, <a href="/A292089/a292089.jpg">Lengths of words in the trajectory of (100)^872</a>
%H A292089 N. J. A. Sloane, <a href="/A291792/a291792.txt">Maple code for A291792, A284119, A291793, A284121, A291794, A291795, A291796, A292089, A292090, A292091, A292092, A292093, A292094.</a>
%H A292089 Shigeru Watanabe, <a href="/A284116/a284116.pdf">Periodicity of Post's normal process of tag</a>, in Jerome Fox, ed., Proceedings of Symposium on Mathematical Theory of Automata, New York, April 1962, Polytechnic Press, Polytechnic Institute of Brooklyn, 1963, pp. 83-99. [Annotated scanned copy]
%e A292089 The following is the analog of columns 3 through 7 of Asveld's Table 1.
%e A292089 1 [171, 6, 56, 59, 138]
%e A292089 2 [166, 6, 56, 59, 133]
%e A292089 3 [11, 6, 16, 17, 10]
%e A292089 4 [154, 6, 56, 59, 121]
%e A292089 5 [105, 0, 0, 31, 24]
%e A292089 6 [14, 518, 28, 85, 215]
%e A292089 7 [57, 6, 38, 41, 36]
%e A292089 8 [68, 518, 42, 85, 333]
%e A292089 9 [173, 0, 0, 49, 38]
%e A292089 10 [1098, 6, 34, 159, 407]
%e A292089 11 [8265, 0, 0, 328, 4429]
%e A292089 12 [720, 6, 34, 93, 343]
%e A292089 13 [1715, 6, 34, 93, 1338]
%e A292089 14 [130, 28, 82, 83, 85]
%e A292089 15 [1979, 6, 20, 215, 720]
%e A292089 16 [2024, 0, 0, 193, 1023]
%e A292089 17 [833, 6, 70, 121, 420]
%e A292089 18 [162, 34, 100, 101, 105]
%e A292089 19 [591, 6, 20, 109, 118]
%e A292089 20 [6124, 0, 0, 357, 2259]
%e A292089 21 [59673, 6, 20, 781, 33530]
%e A292089 22 [748, 0, 0, 150, 328]
%e A292089 23 [11631, 0, 0, 273, 6250]
%e A292089 24 [3200, 6, 56, 261, 1515]
%e A292089 ...
%Y A292089 Cf. A284116, A291067, A291780, A291781.
%Y A292089 Asveld's Table 1 gives data about the behavior of Post's 3-shift tag system {00/1101} applied to the word (100)^n. The first column gives n, the nonzero values in column 2 give A291792, and columns 3 through 7 give A284119, A291793 (or A284121), A291794, A291795, A291796. For the corresponding data for Watanabe's 3-shift tag system {00/1011} applied to (100)^n see A292089, A292090, A292091, A292092, A292093, A292094.
%K A292089 nonn
%O A292089 1,1
%A A292089 _N. J. A. Sloane_, Sep 10 2017
%E A292089 a(8)-(18) from _Lars Blomberg_, Sep 14 2017
%E A292089 a(19) and beyond from _Lars Blomberg_, Apr 20 2018