%I
%S 1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,
%T 0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,
%U 0,0,1,0,0,0,1,0,0,0,0,0,1,0,0,0,1,0
%N 1limiting word of the mapping 00>1000, 10>00, starting with 00.
%C Iterates of the mapping, starting with 00:
%C 00
%C 1000
%C 001000
%C 1000001000
%C 0010001000001000
%C 10000010000010001000001000
%C 001000100000100010000010000010001000001000
%C The 1limiting word is the limit of the nth iterates for n == 1 mod 2.
%C The number of letters (0s and 1s) in the nth iterate is given by 2*F(n+2) for n >= 0, where F = A000045 (Fibonacci numbers), as follows from the observation that this sequence is the {0>00, 1>10}transform of the mapping 0>10, 1>0; see the Example section.
%H Clark Kimberling, <a href="/A288711/b288711.txt">Table of n, a(n) for n = 1..10000</a>
%e The first three nth iterates for n == 1 (mod 3) are
%e 1000
%e 1000001000
%e 10000010000010001000001000
%e Regarding the connection to Fibonacci numbers mentioned in Comments, iterates of the morphism 0>10, 1>0, starting with 0th iterate 0, are
%e 0
%e 10
%e 010
%e 10010
%e 01010010
%e 1001001010010
%e where the length of the nth iterate is F(n+2). The {0>00, 1>10}transform of the iterates is then 00, 1000, 001000, ..., as indicated in Comments.
%t s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
%t w[n_] := StringReplace[w[n  1], {"00" > "1000", "10" > "00"}]
%t Table[w[n], {n, 0, 8}]
%t st = ToCharacterCode[w[9]]  48 (* A288711 *)
%t Flatten[Position[st, 0]] (* A288712 *)
%t Flatten[Position[st, 1]] (* A288713 *)
%t Table[StringLength[w[n]], {n, 0, 20}] (* 2*A000045 *)
%Y Cf. A000045, A288708, A288709.
%K nonn,easy
%O 1
%A _Clark Kimberling_, Jun 16 2017
