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Fixed point of the mapping 00->0010, 01->100, 10->011, starting with 00.
4

%I #8 Apr 06 2020 19:55:49

%S 0,0,1,0,0,1,1,1,0,0,1,0,1,1,1,0,0,1,0,0,1,0,1,1,1,0,0,0,0,1,0,0,1,1,

%T 1,1,0,1,1,0,0,1,0,1,0,0,0,0,1,0,1,1,1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0,

%U 1,0,0,0,1,0,0,1,1,1,1,0,1,1,1,1,0,1

%N Fixed point of the mapping 00->0010, 01->100, 10->011, 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 A289177.

%H Clark Kimberling, <a href="/A289174/b289174.txt">Table of n, a(n) for n = 1..10000</a>

%e The first six iterates of the mapping:

%e 00

%e 0010

%e 0010011

%e 00100111001

%e 00100111001011100

%e 0010011100101110010010110

%t z = 11; (* number of iterates *)

%t s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];

%t w[n_] := StringReplace[w[n - 1], {"00" -> "0010", "01" -> "100", "10" -> "011"}]

%t TableForm[Table[w[n], {n, 0, 10}]]

%t st = ToCharacterCode[w[z]] - 48 (* A289174 *)

%t Flatten[Position[st, 0]] (* A289175 *)

%t Flatten[Position[st, 1]] (* A289176 *)

%t Table[StringLength[w[n]], {n, 0, 20}] (* A289177 *)

%Y Cf. A289175, A289176, A289177.

%K nonn,easy

%O 1

%A _Clark Kimberling_, Jun 29 2017