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%I #34 Aug 11 2018 11:32:01
%S 171,166,11,154,105,14,57,68,173,1098,8265,720,1715,130,1979,2024,833,
%T 162,591,6124,59673,748,11631,3200,1453,13740,2947,2202,15101,1268,
%U 608049,30758,29903,1076,17547,2888,72231,10154,2321,68916,10965,2276,151785,4678
%N Preperiod (or threshold) of orbit of Watanabe's 3-shift tag system {00/1011} applied to the word (100)^n.
%C 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 The empty word is included in the count.
%H Lars Blomberg, <a href="/A292090/b292090.txt">Table of n, a(n) for n = 1..6080</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="/A292090/a292090.png">Graph of the terms</a>
%H Lars Blomberg, <a href="/A292090/a292090_1.png">Graph of preperiod of a(2263)</a>
%H Lars Blomberg, <a href="/A292090/a292090_3.png">Graph of preperiod of a(2540)</a>
%H 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]
%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 The following is the analog of columns 3 through 7 of Asveld's Table 1.
%e 1 [171, 6, 56, 59, 138]
%e 2 [166, 6, 56, 59, 133]
%e 3 [11, 6, 16, 17, 10]
%e 4 [154, 6, 56, 59, 121]
%e 5 [105, 0, 0, 31, 24]
%e 6 [14, 518, 28, 85, 215]
%e 7 [57, 6, 38, 41, 36]
%e 8 [68, 518, 42, 85, 333]
%e 9 [173, 0, 0, 49, 38]
%e 10 [1098, 6, 34, 159, 407]
%e 11 [8265, 0, 0, 328, 4429]
%e 12 [720, 6, 34, 93, 343]
%e 13 [1715, 6, 34, 93, 1338]
%e 14 [130, 28, 82, 83, 85]
%e 15 [1979, 6, 20, 215, 720]
%e 16 [2024, 0, 0, 193, 1023]
%e 17 [833, 6, 70, 121, 420]
%e 18 [162, 34, 100, 101, 105]
%e 19 [591, 6, 20, 109, 118]
%e 20 [6124, 0, 0, 357, 2259]
%e 21 [59673, 6, 20, 781, 33530]
%e 22 [748, 0, 0, 150, 328]
%e 23 [11631, 0, 0, 273, 6250]
%e 24 [3200, 6, 56, 261, 1515]
%e ...
%Y Cf. A284116, A284119, A291067, A291780, A291781.
%Y 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, 291793 (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 nonn
%O 1,1
%A _N. J. A. Sloane_, Sep 10 2017
%E a(25)-(44) from _Lars Blomberg_, Sep 14 2017
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