login
Fixed point of the mapping 00->0110, 10->100, starting with 00.
3

%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