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 A288174 Positions of 0 in A288173; complement of A288175. 3
 1, 2, 4, 5, 6, 8, 9, 11, 12, 13, 14, 16, 17, 19, 20, 21, 23, 24, 26, 27, 28, 30, 31, 32, 34, 35, 37, 38, 39, 41, 42, 44, 45, 46, 47, 49, 50, 52, 53, 54, 56, 57, 59, 60, 61, 62, 64, 65, 67, 68, 69, 70, 72, 73, 75, 76, 77, 79, 80, 82, 83, 84, 85, 87, 88, 90 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Conjecture: lim_{n->infinity} a(n)/n = 1.36..., and if m denotes this number, then -1 < m - a(n)/n < 1 for n >= 1. From Michel Dekking, Feb 23 2020: (Start) Proof of the first part of this conjecture. Let a(0):=0. We write this sequence as the sum of its first differences: a(n) = Sum_{k=0..n-1} a(k+1)-a(k). We know (see A288173) that A288173 can be generated as a decoration delta(t) of the fixed point t of the morphism alpha given by alpha(A) = AB, alpha(B) = AC, alpha(C) = ABB. Here delta is the morphism delta(A) = 001, delta(B) = 0001, delta(C) = 00001. Let e = A288175 be the sequence of positions of 1 in A288173. Note that if we are at the n-th 1, then we have seen e(n)-n zeros. So the position of the (e(n)-n)-th zero is e(n)-1. Let m(n):=e(n)-n. Then a(m(n))/m(n) = (e(n)-1)/m(n) = (e(n)-1)/n * n/(e(n)-n). According to the comments at e = A288175, the first factor in this product converges to 3.7092753596..., and the second to 1/(3.7092753596... - 1). It follows that as n->infinity, a(m(n))/m(n) -> 1.36900369004... . It is easy to see from this that the whole sequence converges, and so a(n)/n -> 1.36900369004... . (End) LINKS Clark Kimberling, Table of n, a(n) for n = 1..10000 MATHEMATICA s = {0, 0}; w[0] = StringJoin[Map[ToString, s]]; w[n_] := StringReplace[w[n - 1], {"00" -> "0010", "1" -> "001"}] Table[w[n], {n, 0, 8}] st = ToCharacterCode[w[11]] - 48 (* A288173 *) Flatten[Position[st, 0]] (* A288174 *) Flatten[Position[st, 1]] (* A288175 *) CROSSREFS Cf. A288173, A288175. Sequence in context: A171599 A328594 A346129 * A280998 A043687 A087118 Adjacent sequences: A288171 A288172 A288173 * A288175 A288176 A288177 KEYWORD nonn,easy AUTHOR Clark Kimberling, Jun 07 2017 STATUS approved

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Last modified March 31 21:40 EDT 2023. Contains 361673 sequences. (Running on oeis4.)