%I #8 Apr 07 2020 22:02:53

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

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

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

%N 2-limiting word of the mapping 00->1000, 10->011, starting with 00.

%C Iterates of the mapping, starting with 00:

%C 00

%C 1000

%C 0111000

%C 0110111000

%C 01011110111000

%C 0011111011110111000

%C 10001111011111011110111000

%C 01110001110111111011111011110111000

%C 011011100011011111110111111011111011110111000

%C The 2-limiting word is the limit of the n-th iterates for n == 2 mod 5.

%C The number of letters (0's and 1's) in the n-th iterate is given by A288243(n), for n >= 0.

%H Clark Kimberling, <a href="/A288858/b288858.txt">Table of n, a(n) for n = 1..10000</a>

%e The first two n-th iterates for n == 2 mod 5: 0111000 and 01110001110111111011111011110111000. (The lengths of the first 10 such iterates are 7, 35, 106, 242, 473, 849, 1442, 2346, 3685, 5642.)

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

%t w[n_] := StringReplace[w[n - 1], {"00" -> "1000", "10" -> "011"}]

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

%t st = ToCharacterCode[w[52]] - 48 (* A288858 *)

%t Flatten[Position[st, 0]] (* A288859 *)

%t Flatten[Position[st, 1]] (* A288860 *)

%t Table[StringLength[w[n]], {n, 0, 30}] (* A288243 *)

%Y Cf. A288226 (0-limiting word), A288855 (1-limiting word), A288861 (3-limiting word), A288864 (4-limiting word), A288859, A288860, A288243.

%K nonn,easy

%O 1

%A _Clark Kimberling_, Jun 23 2017