A289673
Take n-th string over {1,2} in lexicographic order and apply the Post tag system described in A284116 (but adapted to the alphabet {1,2}) just once.
Original entry on oeis.org
-1, 12, 1, 1, 212, 212, 11, 11, 11, 11, 2212, 2212, 2212, 2212, 111, 211, 111, 211, 111, 211, 111, 211, 12212, 22212, 12212, 22212, 12212, 22212, 12212, 22212, 1111, 1211, 2111, 2211, 1111, 1211, 2111, 2211, 1111, 1211, 2111, 2211, 1111, 1211, 2111, 2211, 112212
Offset: 1
The initial words are:
1,2,11,12,21,22,111,112,121,122,211,212,221,222,1111,...
Applying the tag system over {1,2} these become:
-1, 12, 1, 1, 212, 212, 11, 11, 11, 11, 2212, 2212, 2212, 2212, 111, ...
If we were working over {0,1} the initial strings would be:
0,1,00,01,10,11,000,001,010,011,100,101,110,111,0000,...
and applying the tag system over {0,1} described in A284116 these would become:
-1, 01, 0, 0, 101, 101, 00, 00, 00, 00, 1101, 1101, 1101, 1101, 000, ...
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See A291072.
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from itertools import product
A289673_list = [-1 if s == ('1',) else int((''.join(s)+('2212' if s[0] == '2' else '11'))[3:]) for l in range(1,10) for s in product('12',repeat=l)] # Chai Wah Wu, Aug 06 2017
A291069
Largest number of distinct words arising in Watanabe's tag system {00, 0111} applied to a binary word w, over all starting words w of length n.
Original entry on oeis.org
5, 4, 4, 14, 13, 12, 25, 24, 23, 38, 37, 36, 53, 52, 51, 68, 67, 66, 85, 84, 83, 102, 101, 100, 119, 118, 117, 138, 137, 136, 157, 156, 155, 176, 175, 174, 195, 194, 193, 214, 213, 212, 235, 234, 233, 256, 255, 254, 277, 276
Offset: 1
Examples of strings that achieve these records: "1", "10", "000", "1001", "10010", "100100", "1001001".
- 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]
- N. J. A. Sloane, Maple programs that compute first 7 terms for each of A284116, A291067, A291068, A291069
For the 3-shift tag systems {00,1101}, {00, 1011}, {00, 1110}, {00, 0111} see
A284116,
A291067,
A291068,
A291069 respectively (as well as the cross-referenced entries mentioned there).
A291072
Take n-th string over {1,2} in lexicographic order and apply the Watanabe tag system {00, 1011} described in A291067 (but adapted to the alphabet {1,2}) just once.
Original entry on oeis.org
-1, 22, 1, 1, 122, 122, 11, 11, 11, 11, 2122, 2122, 2122, 2122, 111, 211, 111, 211, 111, 211, 111, 211, 12122, 22122, 12122, 22122, 12122, 22122, 12122, 22122, 1111, 1211, 2111, 2211, 1111, 1211, 2111, 2211, 1111, 1211, 2111, 2211, 1111, 1211, 2111, 2211, 112122, 122122, 212122, 222122
Offset: 1
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# First define the mapping by defining the strings T1 and T2:
# Work over the alphabet {1,2}
# 11 / 2212 A284116 This is the "Post Tag System"
T1:="11"; T2:="2212";
# 11 / 2122 A291067 These three are from the Watanabe paper
T1:="11"; T2:="2122";
# 11 / 2221 A291068
T1:="11"; T2:="2221";
# 11 / 1222 A291069
T1:="11"; T2:="1222";
with(StringTools):
# the mapping:
f1:=proc(w) local L, ws, w2; global T1,T2;
ws:=convert(w, string);
if ws="-1" then return("-1"); fi;
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;
# Construct list of words over {1,2} (A007931)
a:= proc(n) local m, r, d; m, r:= n, 0;
while m>0 do d:= irem(m, 2, 'm');
if d=0 then d:=2; m:= m-1 fi;
r:= d, r
od; parse(cat(r))/10
end:
WLIST := [seq(a(n), n=1..100)];
# apply the map once:
# this produces A289673, A291072, A291073, A291074
W2:=map(f1,WLIST);
A291073
Take n-th string over {1,2} in lexicographic order and apply the Watanabe tag system {00, 1110} described in A291068 (but adapted to the alphabet {1,2}) just once.
Original entry on oeis.org
-1, 21, 1, 1, 221, 221, 11, 11, 11, 11, 2221, 2221, 2221, 2221, 111, 211, 111, 211, 111, 211, 111, 211, 12221, 22221, 12221, 22221, 12221, 22221, 12221, 22221, 1111, 1211, 2111, 2211, 1111, 1211, 2111, 2211, 1111, 1211, 2111, 2211, 1111, 1211, 2111
Offset: 1
Showing 1-4 of 4 results.
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