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%I #9 Mar 23 2019 05:35:44
%S 2,4,7,10,12,14,17,19,22,24,27,30,32,35,37,40,42,44,47,50,52,54,57,59,
%T 62,64,67,69,71,74,77,79,82,84,87,90,92,94,97,99,102,104,107,110,112,
%U 115,117,120,122,124,127,130,132,135,137,140,143,145,147,150
%N Positions of 1 in A285966; complement of A285967.
%C Conjecture: a(n)/n -> 5/2.
%C From _Michel Dekking_, Mar 15 2019: (Start)
%C Proof of this conjecture: it is equivalent to prove that the frequency of 1 in A285966 exists and equals 2/5.
%C This follows from my characterization of A285966 as a morphic sequence.
%C The incidence matrix of that morphism has Perron-Frobenius eigenvalue 2, with right eigenvector (1,1,2,1). It follows that the letters 2 and 4 in the fixed point of the morphism have both frequency 1/5. As these are exactly the letters that are mapped to 1 in A285966, the letter 1 has frequency 2/5 in A285966.
%C (End)
%H Clark Kimberling, <a href="/A285968/b285968.txt">Table of n, a(n) for n = 1..10000</a>
%e As a word, A285966 = 01010010010101001010010..., in which 1 is in positions 2,4,7,10,12,...
%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 9] (* Thue-Morse, A010060 *)
%t w = StringJoin[Map[ToString, s]]
%t w1 = StringReplace[w, {"11" -> "1"}]
%t st = ToCharacterCode[w1] - 48 (* A285966 *)
%t Flatten[Position[st, 0]] (* A285967 *)
%t Flatten[Position[st, 1]] (* A285968 *)
%Y Cf. A010060, A285966, A285967.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, May 06 2017
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