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%I #10 Jun 20 2020 22:53:00
%S 0,1,1,0,0,0,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,0,1,1,0,1,1,0,0,0,1,1,0,
%T 1,1,0,0,1,1,0,0,0,1,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,0,1,1,0,1,1,0,0,1,
%U 1,0,0,0,1,1,0,0,0,1,1,0,1,1,0,0,0,1
%N Fixed point of the mapping 00->0110, 10->100, starting with 00.
%C Iterates, starting with 00:
%C 00
%C 0110
%C 01100
%C 011000
%C 011000110
%C 0110001101100
%C 011000110110011000
%C 01100011011001100011000110
%C Conjecture: the number of letters (0's and 1's) in the n-th iterate is given by A288429(n).
%H Clark Kimberling, <a href="/A288694/b288694.txt">Table of n, a(n) for n = 1..10000</a>
%t s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
%t w[n_] := StringReplace[w[n - 1], {"00" -> "0110", "10" -> "100"}]
%t Table[w[n], {n, 0, 8}]
%t st = ToCharacterCode[w[11]] - 48 (* A288694 *)
%t Flatten[Position[st, 0]] (* A288695 *)
%t Flatten[Position[st, 1]] (* A288696 *)
%t Table[StringLength[w[n]], {n, 1, 35}] (* A288429 conjectured *)
%Y Cf. A288695, A288696, A288429.
%K nonn,easy
%O 1
%A _Clark Kimberling_, Jun 16 2017
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