A290742 Orbit of word "1" under the 3-shift tag system over the alphabet {1,2} defined in the Comments.

1, 12, 112, 2112, 21221222, 212221221222, 2212212221221222, 22122212212221221222, 222122122212212221221222, 1221222122122212212221221222, 12221221222122122212212222112, 212212221221222122122221122112, 2122212212221221222211221121221222

Offset: 1

Author

N. J. A. Sloane, Aug 11 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 three letters, or if w begins with 2, w' is obtained by appending 1221222 to w and deleting the first three letters.

This is a 3-shift version of a 5-shift tag system studied in [De Mol, p. 307].

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..322
• 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.

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[4..L]; end;
# and apply f1 repeatedly to "1"

Python

from itertools import islice
def agen(w="1"):
    while True:
        yield int(w)
        w += ("2112" if w[0] == "1" else "1221222")
        w = w[3:]
print(list(islice(agen(), 13))) # Michael S. Branicky, Mar 15 2022