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1, 2, 4, 6, 7, 9, 10, 11, 13, 15, 16, 17, 19, 20, 22, 24, 25, 27, 28, 29, 31, 32, 34, 36, 37, 38, 40, 42, 43, 45, 46, 47, 49, 51, 52, 53, 55, 56, 58, 60, 61, 62, 64, 66, 67, 69, 70, 71, 73, 74, 76, 78, 79, 81, 82, 83, 85, 87, 88, 89, 91, 92, 94, 96, 97, 99
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OFFSET
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1,2
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COMMENTS
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Conjecture: 3n/2 - a(n) is in {0, 1/2, 1} for n >= 1.
Proof of the conjecture: Let t=A010060 be the Thue-Morse sequence. Every pair t(2n-1),t(2n) is either 01 or 10. Since 01 and 10 map to 110 and 101 under the transform, which both have length 3, it follows that a(2n+1) = 3n+1, and a(2n) = 3n if t(2n)=0, a(2n) = 3n-1 if t(2n)=1 for n=1,2,..., and so certainly 3n/2 - a(n) is 0, 1/2 or 1. - Michel Dekking, Jan 05 2018
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LINKS
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FORMULA
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EXAMPLE
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As a word, A285952 = 110101101110101..., in which 1 is in positions 1,2,4,6,7,9,...
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MATHEMATICA
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s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 7] (* Thue-Morse, A010060 *)
w = StringJoin[Map[ToString, s]]
w1 = StringReplace[w, {"0" -> "1", "1" -> "10"}] (* A285952, word *)
st = ToCharacterCode[w1] - 48 (* A285952, sequence *)
Flatten[Position[st, 0]] (* A285953 *)
Flatten[Position[st, 1]] (* A285954 *)
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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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