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 A288711 1-limiting word of the mapping 00->1000, 10->00, starting with 00. 5
 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, 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, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1 COMMENTS Iterates of the mapping, starting with 00: 00 1000 001000 1000001000 0010001000001000 10000010000010001000001000 001000100000100010000010000010001000001000 The 1-limiting word is the limit of the n-th iterates for n == 1 mod 2. The number of letters (0's and 1's) in the n-th 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. LINKS Clark Kimberling, Table of n, a(n) for n = 1..10000 EXAMPLE The first three n-th iterates for n == 1 (mod 3) are 1000 1000001000 10000010000010001000001000 Regarding the connection to Fibonacci numbers mentioned in Comments, iterates of the morphism 0->10, 1->0, starting with 0th iterate 0, are 0 10 010 10010 01010010 1001001010010 where the length of the n-th iterate is F(n+2). The {0->00, 1->10}-transform of the iterates is then 00, 1000, 001000, ..., as indicated in Comments. MATHEMATICA s = {0, 0}; w[0] = StringJoin[Map[ToString, s]]; w[n_] := StringReplace[w[n - 1], {"00" -> "1000", "10" -> "00"}] Table[w[n], {n, 0, 8}] st = ToCharacterCode[w[9]] - 48   (* A288711 *) Flatten[Position[st, 0]]  (* A288712 *) Flatten[Position[st, 1]]  (* A288713 *) Table[StringLength[w[n]], {n, 0, 20}] (* 2*A000045 *) CROSSREFS Cf. A000045, A288708, A288709. Sequence in context: A014189 A319691 A079979 * A347312 A089010 A162289 Adjacent sequences:  A288708 A288709 A288710 * A288712 A288713 A288714 KEYWORD nonn,easy AUTHOR Clark Kimberling, Jun 16 2017 STATUS approved

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Last modified September 28 19:16 EDT 2021. Contains 347717 sequences. (Running on oeis4.)