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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.

%I A291072 #8 Dec 12 2023 13:39:57
%S A291072 -1,22,1,1,122,122,11,11,11,11,2122,2122,2122,2122,111,211,111,211,
%T A291072 111,211,111,211,12122,22122,12122,22122,12122,22122,12122,22122,1111,
%U A291072 1211,2111,2211,1111,1211,2111,2211,1111,1211,2111,2211,1111,1211,2111,2211,112122,122122,212122,222122
%N 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.
%p A291072 # First define the mapping by defining the strings T1 and T2:
%p A291072 # Work over the alphabet {1,2}
%p A291072 # 11 / 2212 A284116 This is the "Post Tag System"
%p A291072 T1:="11"; T2:="2212";
%p A291072 # 11 / 2122 A291067 These three are from the Watanabe paper
%p A291072 T1:="11"; T2:="2122";
%p A291072 # 11 / 2221 A291068
%p A291072 T1:="11"; T2:="2221";
%p A291072 # 11 / 1222 A291069
%p A291072 T1:="11"; T2:="1222";
%p A291072 with(StringTools):
%p A291072 # the mapping:
%p A291072 f1:=proc(w) local L, ws, w2; global T1,T2;
%p A291072 ws:=convert(w, string);
%p A291072 if ws="-1" then return("-1"); fi;
%p A291072 if ws[1]="1" then w2:=Join([ws, T1], ""); else w2:=Join([ws, T2], "");  fi;
%p A291072 L:=length(w2); if L <= 3 then return("-1"); fi;
%p A291072 w2[4..L]; end;
%p A291072 # Construct list of words over {1,2} (A007931)
%p A291072 a:= proc(n) local m, r, d; m, r:= n, 0;
%p A291072       while m>0 do d:= irem(m, 2, 'm');
%p A291072         if d=0 then d:=2; m:= m-1 fi;
%p A291072         r:= d, r
%p A291072       od; parse(cat(r))/10
%p A291072     end:
%p A291072 WLIST := [seq(a(n), n=1..100)];
%p A291072 # apply the map once:
%p A291072 # this produces A289673, A291072, A291073, A291074
%p A291072 W2:=map(f1,WLIST);
%Y A291072 Cf. A007931, A284116, A291067, A291068, A291069.
%Y A291072 Cf. also A289673, A291073, A291074.
%K A291072 sign
%O A291072 1,2
%A A291072 _N. J. A. Sloane_, Aug 18 2017