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%I #17 Apr 07 2020 21:11:17
%S 0,0,0,0,0,1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,0,0,0,0,0,0,
%T 1,1,0,0,0,0,0,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0,0,0,1,1,0,0,1,1,0,0,1,
%U 0,0,0,0,1,1,0,0,1,1,0,0,1,0,0,0,0,1
%N 0-limiting word of the mapping 00->0110, 10->000, starting with 00.
%C Iterates of the mapping, starting with 00:
%C 00
%C 0110
%C 01000
%C 00000110
%C 0110011001000
%C 010000100000000110
%C 00000110000001100110011001000
%C 01100110010000110011001000010000100000000110
%C The 0-limiting word is the limit of the n-th iterates for n == 0 mod 3.
%C Conjecture: the number of letters (0's and 1's) in the n-th iterate is given by A288468(n).
%H Clark Kimberling, <a href="/A288665/b288665.txt">Table of n, a(n) for n = 1..10000</a>
%e The first three n-th iterates for n == 0 mod 3 are
%e 00
%e 00000110
%e 00000110000001100110011001000
%t s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
%t w[n_] := StringReplace[w[n - 1], {"00" -> "0110", "10" -> "000"}]
%t Table[w[n], {n, 0, 8}]
%t st = ToCharacterCode[w[9]] - 48 (* A288665 *)
%t Flatten[Position[st, 0]] (* A288666 *)
%t Flatten[Position[st, 1]] (* A288667 *)
%t Table[StringLength[w[n]], {n, 0, 20}] (* A288668 conjectured *)
%Y Cf. A288666, A288667, A288668, A299670 (1-limiting word), A299673 (2-limiting word).
%K nonn,easy
%O 1
%A _Clark Kimberling_, Jun 15 2017
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