A292092 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.

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, 46, 64, 64, 64, 92, 74, 34, 118, 66, 88, 52, 0, 0, 34, 268, 42, 34, 0, 46, 30, 92, 0, 16, 34, 76, 76, 34, 34, 38, 110, 20, 64, 92, 46, 56, 0, 46, 76, 0, 74, 0, 88, 0

Offset

1,1

Comments

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.

The empty word is included in the count.

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

Links

Lars Blomberg, Table of n, a(n) for n = 1..6080

Peter R. J. Asveld, On a Post's System of Tag. Bulletin of the EATCS 36 (1988), 96-102.

Shigeru Watanabe, Periodicity of Post's normal process of tag, 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]

Example

The following is the analog of columns 3 through 7 of Asveld's Table 1.
1 [171, 6, 56, 59, 138]
2 [166, 6, 56, 59, 133]
3 [11, 6, 16, 17, 10]
4 [154, 6, 56, 59, 121]
5 [105, 0, 0, 31, 24]
6 [14, 518, 28, 85, 215]
7 [57, 6, 38, 41, 36]
8 [68, 518, 42, 85, 333]
9 [173, 0, 0, 49, 38]
10 [1098, 6, 34, 159, 407]
11 [8265, 0, 0, 328, 4429]
12 [720, 6, 34, 93, 343]
13 [1715, 6, 34, 93, 1338]
14 [130, 28, 82, 83, 85]
15 [1979, 6, 20, 215, 720]
16 [2024, 0, 0, 193, 1023]
17 [833, 6, 70, 121, 420]
18 [162, 34, 100, 101, 105]
19 [591, 6, 20, 109, 118]
20 [6124, 0, 0, 357, 2259]
21 [59673, 6, 20, 781, 33530]
22 [748, 0, 0, 150, 328]
23 [11631, 0, 0, 273, 6250]
24 [3200, 6, 56, 261, 1515]
...

Crossrefs

Cf. A284116, A291067, A291780, A291781.

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.

Sequence in context: A278826 A259537 A359604 * A003904 A262148 A008942

Adjacent sequences: A292089 A292090 A292091 * A292093 A292094 A292095

Keyword

nonn

Author

N. J. A. Sloane, Sep 10 2017

Extensions

a(25)-(68) from Lars Blomberg, Sep 14 2017

Status

approved