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%I #7 Apr 06 2020 19:33:24
%S 0,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0,1,0,0,1,0,1,1,0,0,1,0,1,0,
%T 0,1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1,1,0,0,1,0,0,1,0,1,1,0,0,1,
%U 0,1,0,0,1,0,1,1,0,0,1,0,1,0,0,1,0,1
%N Fixed point of the mapping 00->0101, 1->10, starting with 00.
%C Conjecture: the number of letters (0's and 1's) in the n-th iterate is given by A288465(n).
%H Clark Kimberling, <a href="/A288462/b288462.txt">Table of n, a(n) for n = 1..10000</a>
%e Iterates, starting with 00:
%e 00
%e 0101
%e 010010
%e 0100101100
%e 010010110010100101
%e 01001011001010010110010010110010
%t s = {0, 0}; w[0] = StringJoin[Map[ToString, s]];
%t w[n_] := StringReplace[w[n - 1], {"00" -> "0101", "1" -> "10"}]
%t Table[w[n], {n, 0, 8}]
%t st = ToCharacterCode[w[11]] - 48 (* A288462 *)
%t Flatten[Position[st, 0]] (* A288463 *)
%t Flatten[Position[st, 1]] (* A288464 *)
%t Table[StringLength[w[n]], {n, 1, 35}] (* A288465 conjectured *)
%Y Cf. A288463, A288464, A288465.
%K nonn,easy
%O 1
%A _Clark Kimberling_, Jun 11 2017
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