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%I #6 Apr 07 2020 21:16:11
%S 0,0,1,1,0,1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,
%T 1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,
%U 0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1
%N Fixed point of the mapping 00->001, 1->101, 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 A132045.
%H Clark Kimberling, <a href="/A286807/b286807.txt">Table of n, a(n) for n = 1..10000</a>
%e Iterates, starting with 00:
%e 00
%e 001
%e 001101
%e 0011011010101
%e 0011011010101101010101010101
%e 00110110101011010101010101011010101010101010101010101010101
%t s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
%t w[n_] := StringReplace[w[n - 1], {"00" -> "001", "1" -> "101"}]
%t Table[w[n], {n, 0, 8}]
%t st = ToCharacterCode[w[11]] - 48 (* A286807 *)
%t Flatten[Position[st, 0]] (* A286808 *)
%t Flatten[Position[st, 1]] (* A286809 *)
%t Table[StringLength[w[n]], {n, 1, 35}] (* A132045 conjectured *)
%Y Cf. A286808, A286809, A132045.
%K nonn,easy
%O 1
%A _Clark Kimberling_, Jun 05 2017
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