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%I #11 Apr 07 2020 21:03:28
%S 1,0,0,0,0,1,0,0,0,1,0,0,1,0,1,0,0,0,0,1,0,1,0,0,0,0,0,1,0,0,0,1,0,0,
%T 1,0,1,0,0,0,1,0,0,0,0,1,0,0,1,0,1,0,0,0,0,0,1,0,1,0,0,0,0,1,0,0,0,1,
%U 0,0,1,0,1,0,0,0,0,1,0,1,0,0,0,0,0,1
%N 1-limiting word of the mapping 00->1000, 10->010, starting with 00.
%C Iterates of the mapping, starting with 00:
%C 00
%C 1000
%C 0101000
%C 00100101000
%C 100001000100101000
%C 01010000010100001000100101000
%C 00100101000100001001010000010100001000100101000
%C The 1-limiting word is the limit of the n-th iterates for n == 1 mod 3.
%C The number of letters (0's and 1's) in the n-th iterate is given by A288219(n), for n >= 0.
%H Clark Kimberling, <a href="/A288220/b288220.txt">Table of n, a(n) for n = 1..10000</a>
%e The first two n-th iterates for n == 1 mod 3:
%e 1000
%e 100001000100101000
%t s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
%t w[n_] := StringReplace[w[n - 1], {"00" -> "1000", "10" -> "010"}]
%t Table[w[n], {n, 0, 8}]
%t st = ToCharacterCode[w[22]] - 48 (* A288220 *)
%t Flatten[Position[st, 0]] (* A288221 *)
%t Flatten[Position[st, 1]] (* A288222 *)
%t Table[StringLength[w[n]], {n, 0, 30}] (* A288219 *)
%Y Cf. A288216 (0-limiting word), A288221, A288222, A288219, A288223 (2-limiting word).
%K nonn,easy
%O 1
%A _Clark Kimberling_, Jun 19 2017
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