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A291793 Period of orbit of Post's tag system applied to the word (100)^n (version 2), or -1 if the orbit increases without limit. 9
2, 6, 6, 6, 0, 10, 28, 6, 10, 6, 6, 6, 0, 0, 6, 28, 10, 6, 10, 6, 6, 0, 6, 6, 0, 6, 6, 6, 6, 6, 6, 52, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 28, 6, 0, 0, 28, 6, 6, 6, 6, 6, 0, 6, 6, 6, 10, 6, 6, 6, 6, 0, 6, 0, 6, 6, 6, 6, 0, 6, 6, 6, 0, 6, 6, 6, 0, 10, 0, 10, 6, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

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.

The empty word is included in the count.

Here, following Asveld, a(n)=0 if the orbit ends at the empty word. On the other hand, Shallit defines a(n) to be 1 if that happens, which gives a different sequence, A284121.

From A.H.M. Smeets, Jul 16 2020: (Start)

In general a tag as defined by Emil Leon Post, is given by a 4-tuple (Sigma,AF,n,w0), where Sigma is some (nonempty) alphabet, AF is the associated function (sometimes also called set of production rules) AF: Sigma -> Sigma*, n is the deletion number and w0 the initial string.

Here, the period lengths a(n) refer to the tags ({0,1},{(0,00),(1,1101)},3,100^n).

a(n) is an even number. Proof: for each cycle the number of associations (productions) 0 -> 00 must equal the number of associations (productions) 1 -> 1101 applied within a cycle. (End)

LINKS

Lars Blomberg, Table of n, a(n) for n = 1..6075 (corrected for n=165 by A.H.M. Smeets)

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

Lars Blomberg, Histogram over non-zero terms

Emil L. Post, Formal reductions of the general combinatorial decision problem., American Journal of Mathematics, Vol. 65, No. 2 (Apr., 1943), pp. 197-215.

Eric Weisstein's World of Mathematics, Tag System

EXAMPLE

For n = 2 the orbit of (100)^2 = 100100 consists of a preperiod of length 15, followed by a periodic portion of length 6.

PROG

(Python)

def step(w):

    i = 0

    while w[0] != alfabet[i]:

        i = i+1

    w = w+suffix[i]

    return w[n:len(w)]

alfabet, suffix, n, ws, w0, m = "01", ["00", "1101"], 3, "100", "", 0

while m < 83:

    w0, m = w0+ws, m+1

    w, ww, i, a = w0, w0, 0, 0

    while w != "" and a == 0:

        w, i = step(w), i+1

        if i%1000 == 0:

            ww = w

        else:

            if w == ww or w == "":

                if w != "":

                    a = i%1000

                print(m, a) # A.H.M. Smeets, Jul 16 2020

CROSSREFS

Cf. A284116, A284119, A291792, A284121, A336287, A336327.

Sequence in context: A065486 A069806 A123945 * A284121 A198102 A097466

Adjacent sequences:  A291790 A291791 A291792 * A291794 A291795 A291796

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Sep 04 2017, based on Jeffrey Shallit's A284121.

EXTENSIONS

a(50)-a(83) from Lars Blomberg, Sep 08 2017

STATUS

approved

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Last modified March 4 23:37 EST 2021. Contains 341812 sequences. (Running on oeis4.)