A293945 Orbit of word "2" under De Mol's 5-shift tag system over the alphabet {1,2} defined in the Comments.

2, 222, 21222, 1221222, 222112, 21221222, 2221221222, 212221221222, 12212221221222, 2212212222112, 122221121221222, 11212212222112, 2122221122112, 211221121221222, 11212212221221222, 2122212212222112, 122122221121221222, 22211212212222112, 2122122221121221222, 222211212212221221222

Offset: 1

Author

N. J. A. Sloane, Oct 23 2017

Keywords

• nonn

Comments

This tag system maps a word w over {1,2} to w', where if w begins with 1, w' is obtained by appending 2112 to w and deleting the first five letters, or if w begins with 2, w' is obtained by appending 1221222 to w and deleting the first five letters.

It can be shown that under this tag system every word different from "1" has an infinite orbit [De Mol, p. 307].

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..1025
• Liesbeth De Mol, Tracing unsolvability. A historical, mathematical and philosophical analysis with a special focus on tag systems, Ph.D. Thesis, Universiteit Gent. See page 307.

Crossrefs

Cf. A284116, A290741, A290742.

Programs

Maple

with(StringTools);
f1:=proc(w) local L, t2, t1, ws, w2;
t1:="2112"; t2:="1221222"; ws:=convert(w, string);
if ws[1]="1" then w2:=Join([ws, t1], ""); else w2:=Join([ws, t2], ""); fi;
L:=length(w2); if L <= 3 then return(-1); fi;
w2[6..L]; end;
# and apply f1 repeatedly to "2"

Python

from itertools import islice
def agen(w="2"):
    while True:
        yield int(w)
        w += ("2112" if w[0] == "1" else "1221222")
        w = w[5:]
print(list(islice(agen(), 20))) # Michael S. Branicky, Jan 06 2025