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%I #6 Apr 07 2020 21:17:35
%S 1,0,0,0,1,0,1,1,0,1,1,0,0,0,1,0,0,0,1,0,1,1,0,1,1,0,1,1,0,1,1,0,0,0,
%T 1,0,0,0,1,0,1,1,0,1,1,0,0,0,1,0,0,0,1,0,1,1,0,1,1,0,1,1,0,1,1,0,1,1,
%U 0,1,1,0,0,0,1,0,0,0,1,0,1,1,0,1,1,0
%N 1-limiting word of the mapping 00->1000, 10->01, starting with 00.
%C Iterates of the mapping, starting with 00:
%C 00
%C 1000
%C 011000
%C 01011000
%C 0011011000
%C 10001011011000
%C 011000011011011000
%C 0101100001011011011000
%C 00110110000011011011011000
%C 1000101101100010001011011011011000
%C The 1-limiting word is the limit of the n-th iterates for n == 1 mod 4. Conjecture: the number of letters (0's and 1's) in the n-th iterate is given by A288732(n), for n >= 0.
%H Clark Kimberling, <a href="/A288733/b288733.txt">Table of n, a(n) for n = 1..10000</a>
%e The first three n-th iterates for n == 1 mod 3 are
%e 1000
%e 10001011011000
%e 1000101101100010001011011011011000
%t s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
%t w[n_] := StringReplace[w[n - 1], {"00" -> "1000", "10" -> "01"}]
%t Table[w[n], {n, 0, 8}]
%t st = ToCharacterCode[w[21]] - 48 (* A288733 *)
%t Flatten[Position[st, 0]] (* A288734 *)
%t Flatten[Position[st, 1]] (* A288735 *)
%t Table[StringLength[w[n]], {n, 0, 20}] (* A288732 *)
%Y Cf. A288729 (0-limiting word), A288734, A288735, A288732, A288736 (2-limiting word), A288741 (3-limiting word).
%K nonn,easy
%O 1
%A _Clark Kimberling_, Jun 16 2017
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