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%I #7 Apr 07 2020 21:50:25
%S 1,0,0,0,1,1,1,1,0,1,1,1,1,1,0,1,1,1,1,0,1,1,1,0,0,0,0,1,1,1,1,1,1,1,
%T 1,0,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,0,1,1,1,1,1,
%U 1,0,1,1,1,1,1,0,1,1,1,1,0,1,1,1,0,0
%N 1-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 1-limiting word is the limit of the n-th iterates for n == 1 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="/A288855/b288855.txt">Table of n, a(n) for n = 1..10000</a>
%e The first two n-th iterates for n == 1 mod 5: 1000 and 10001111011111011110111000. (The lengths of the first 10 such iterates are 4, 26, 87, 208, 417, 759, 1302, 2136, 3376, 5192.)
%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[51]] - 48 (* A288855 *)
%t Flatten[Position[st, 0]] (* A288856 *)
%t Flatten[Position[st, 1]] (* A288857 *)
%t Table[StringLength[w[n]], {n, 0, 30}] (* A288243 *)
%Y Cf. A288226 (0-limiting word), A288858 (2-limiting word), A288861 (3-limiting word), A288864 (4-limiting word), A288856, A288857, A288243.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, Jun 22 2017
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