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0-limiting word of the mapping 00->0110, 10->000, starting with 00.
6

%I #21 Jul 27 2024 09:33:49

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

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

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

%N 0-limiting word of the mapping 00->0110, 10->000, starting with 00.

%C Iterates of the mapping, starting with 00:

%C 00

%C 0110

%C 01000

%C 00000110

%C 0110011001000

%C 010000100000000110

%C 00000110000001100110011001000

%C 01100110010000110011001000010000100000000110

%C The 0-limiting word is the limit of the n-th iterates for n == 0 mod 3.

%C Conjecture: the number of letters (0's and 1's) in the n-th iterate is given by A288668(n).

%H Clark Kimberling, <a href="/A288665/b288665.txt">Table of n, a(n) for n = 1..10000</a>

%e The first three n-th iterates for n == 0 mod 3 are

%e 00

%e 00000110

%e 00000110000001100110011001000

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

%t w[n_] := StringReplace[w[n - 1], {"00" -> "0110", "10" -> "000"}]

%t Table[w[n], {n, 0, 8}]

%t st = ToCharacterCode[w[9]] - 48 (* A288665 *)

%t Flatten[Position[st, 0]] (* A288666 *)

%t Flatten[Position[st, 1]] (* A288667 *)

%t Table[StringLength[w[n]], {n, 0, 20}] (* A288668 conjectured *)

%Y Cf. A288666, A288667, A288668, A299670 (1-limiting word), A299673 (2-limiting word).

%K nonn,easy

%O 1

%A _Clark Kimberling_, Jun 15 2017