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Fixed point of the mapping 00->001, 1->101, starting with 00.
3

%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