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%I #8 Apr 06 2020 20:17:06
%S 1,0,0,0,1,0,0,1,1,0,0,0,1,0,0,0,0,0,1,0,0,1,0,1,0,0,1,1,0,0,0,1,0,0,
%T 0,1,0,0,1,1,0,0,0,0,0,1,1,0,0,0,1,0,0,0,0,0,1,1,0,0,0,0,0,1,0,0,0,1,
%U 0,0,1,0,1,0,0,1,1,0,0,0,1,0,0,0,1,0
%N 1-limiting word of the mapping 00->1000, 10->001, starting with 00.
%C Iterates of the mapping, starting with 00:
%C 00
%C 1000
%C 0011000
%C 100010011000
%C 0011000001010011000
%C 1000100110001000001001010011000
%C 0011000001010011000001100010000010001001010011000
%C The 1-limiting word is the limit of the n-th iterates for odd n.
%C Conjecture: the number of letters (0's and 1's) in the n-th iterate is given by A287525(n), for n >= 0.
%H Clark Kimberling, <a href="/A287530/b287530.txt">Table of n, a(n) for n = 1..10000</a>
%e The first three n-th iterates for even n:
%e 1000
%e 100010011000
%e 1000100110001000001001010011000
%t s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
%t w[n_] := StringReplace[w[n - 1], {"00" -> "1000", "10" -> "001"}]
%t Table[w[n], {n, 0, 8}]
%t st = ToCharacterCode[w[23]] - 48 (* A287530 *)
%t Flatten[Position[st, 0]] (* A287531 *)
%t Flatten[Position[st, 1]] (* A287551 *)
%t Table[StringLength[w[n]], {n, 0, 30}] (* A287525 *)
%Y Cf. A287382 (0-limiting word), A287531, A287551, A287525.
%K nonn,easy
%O 1
%A _Clark Kimberling_, Jun 18 2017
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