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Conjecture: the number of letters (0s 0's and 1s1's) in the n-th iterate of the mapping is given by A289004.

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allocated for Clark KimberlingFixed point of the mapping 00->0010, 01->100, 10->010, starting with 00.

0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0

1

Conjecture: the number of letters (0s and 1s) in the n-th iterate of the mapping is given by A289004.

Clark Kimberling, <a href="/A289239/b289239.txt">Table of n, a(n) for n = 1..10000</a>

The first six iterates of the mapping:

00

0010

0010010

00100101000

001001010010000100

0010010100100001001000101000010

z = 11; (* number of iterates *)

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

w[n_] := StringReplace[w[n - 1], {"00" -> "0010", "01" -> "100", "10" -> "010"}]

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

st = ToCharacterCode[w[z]] - 48 (* A289239 *)

Flatten[Position[st, 0]] (* A289240 *)

Flatten[Position[st, 1]] (* A289241 *)

Table[StringLength[w[n]], {n, 0, 20}] (* A289004 conjectured *)

Cf. A289240, A289241, A289004.

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nonn,easy

Clark Kimberling, Jun 29 2017

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allocated for Clark Kimberling

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