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1, 2, 4, 5, 6, 8, 9, 10, 11, 12, 14, 15, 16, 18, 19, 20, 21, 22, 24, 25, 26, 27, 28, 30, 31, 32, 34, 35, 36, 37, 38, 40, 41, 42, 44, 45, 46, 47, 48, 50, 51, 52, 53, 54, 56, 57, 58, 60, 61, 62, 63, 64, 66, 67, 68, 69, 70, 72, 73, 74, 76, 77, 78, 79, 80, 82
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graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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Conjecture: a(n)/n -> -1 + sqrt(5), and if m denotes this number, then -1 < m - a(n)/n < 1 for n >= 1.
Proof of this conjecture. One follows the same strategy as in the proof of the conjecture in A288709. We have a(n)/n -> -1 + sqrt(5) if and only if n/a(n) -> (1+sqrt(5))/4, which is the frequency of 0's in A288707. If alpha = (3-sqrt(5))/2 is the frequency of 1's in y = A189661 = 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, ... then (2*(1-alpha)+alpha)/2 is the frequency of 0's in A288707, since every 0 in y produces 2 zeros in A288707, and every 1 just 1 zero (and one 1). One computes (2*(1-alpha)+alpha)/2 = (1+sqrt(5))/4. The bounds follow from the exponential convergence of the frequencies of 0 and 1 in y. - Michel Dekking, Oct 19 2018
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LINKS
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MATHEMATICA
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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[10]] - 48 (* A288707 *)
Flatten[Position[st, 0]] (* A288708 *)
Flatten[Position[st, 1]] (* A288709 *)
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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