A292092 - OEIS

%I #20 Aug 11 2018 11:32:08

%S 56,56,16,56,0,28,38,42,0,34,0,34,34,82,20,0,70,100,20,0,20,0,0,56,0,

%T 46,64,64,64,92,74,34,118,66,88,52,0,0,34,268,42,34,0,46,30,92,0,16,

%U 34,76,76,34,34,38,110,20,64,92,46,56,0,46,76,0,74,0,88,0

%N Consider Watanabe's 3-shift tag system {00/1011} applied to the word (100)^n; a(n) = length of first word we see that is in the cycle, if the orbit cycles, or 0 if the orbit reaches the empty string, or -1 if the orbit is unbounded.

%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.

%C Following Asveld we set a(n)=0 if the orbit ends at the empty word.

%H Lars Blomberg, <a href="/A292092/b292092.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 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 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, 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)-(68) from _Lars Blomberg_, Sep 14 2017