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%I #24 Feb 16 2021 02:02:57
%S 1,0,1,0,1,1,0,1,0,1,1,1,0,1,0,1,1,0,1,0,1,1,1,1,0,1,0,1,1,0,1,0,1,1,
%T 1,0,1,0,1,1,0,1,0,1,1,1,1,1,0,1,0,1,1,0,1,0,1,1,1,0,1,0,1,1,0,1,0,1,
%U 1,1,1,0,1,0,1,1,0,1,0,1,1,1,0,1,0,1
%N Fixed point of the mapping 00->1000, 10->10101, starting with 00.
%C Conjecture: The number of letters (0's and 1's) in the n-th iterate of the mapping is given by A123720.
%C The first five iterates of the mapping:
%C 00
%C 1000
%C 101011000
%C 10101101011101011000
%C 1010110101110101101011110101101011101011000
%C No iterate is an initial subword of its successor.
%H Clark Kimberling, <a href="/A288932/b288932.txt">Table of n, a(n) for n = 1..10000</a>
%H Michel Dekking, <a href="/A288932/a288932.pdf">A288932 A308185 connection</a>
%F a(n+1) = A308185(n). - _Michel Dekking_, Feb 15 2021
%t s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
%t w[n_] := StringReplace[w[n - 1], {"00" -> "1000", "10" -> "10101"}]
%t Table[w[n], {n, 0, 8}]
%t st = ToCharacterCode[w[11]] - 48 (* A288932 *)
%t Flatten[Position[st, 0]] (* A288933 *)
%t Flatten[Position[st, 1]] (* A288934 *)
%t Table[StringLength[w[n]], {n, 1, 35}] (* A123720 conjectured *)
%Y Cf. A288933, A288934, A123720, A308185.
%K nonn,easy
%O 1
%A _Clark Kimberling_, Jun 25 2017
%E Corrected by _Clark Kimberling_, Feb 13 2021
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