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%I #22 Sep 18 2017 03:09:23
%S 2,3,5,9,11,12,14,17,19,20,23,27,29,30,33,36,38,39,41,45,47,48,50,53,
%T 55,56,59,62,64,65,67,71,73,74,77,81,83,84,86,89,91,92,95,99,101,102,
%U 105,108,110,111,113,117,119,120,123,127,129,130,132,135,137
%N Positions of 1 in A284622; complement of A284623.
%C a(n) - a(n-1) is in {1,2,3,4} for n >= 2.
%C Conjecture: a(n)/n -> 9/4.
%C From _Michel Dekking_, Aug 26 2017: (Start)
%C Proof of the conjecture. Let x = A010060 be the Thue Morse sequence, and let y = A284622 be the [0011->0]-transform of x. Let a = A284626 be the positions of 1 in y. There are 3 steps in the proof.
%C Step 1. It is easily verified that a(n)/n -> 9/4 if and only if f(1,a) = 4/9, where in general f(w,z) denotes the frequency of a word w in the infinite sequence z, if it exists.
%C Step 2. One has f(0011,x) = 1/12. It is well-known that the frequencies of words in any fixed point of a primitive morphism exist. This is usually proved by Perron-Frobenius theory. For a quick proof see the paper "On the Thue-Morse measure".
%C Step 3. Let k(n) be the number of 1's in x(1)...x(n), and m(n) the number of 0011's in x(1)...x(n). Then the number of 1's in y(1)...y(n-3m(n)) is equal to k(n)-2m(n). But we know by Step 2 that m(n)/n -> 1/12, and obviously k(n)/n -> 1/2. So f(1,y) is equal to ((1/2 - 2/12)/(1 - 3/12) = 4/9. (End)
%H Clark Kimberling, <a href="/A284626/b284626.txt">Table of n, a(n) for n = 1..10000</a>
%H Michel Dekking,<a href="https://dml.cz/bitstream/handle/10338.dmlcz/701974/ActaCarolinae_033-1992-2_6.pdf">On the Thue-Morse measure</a>, Acta Universitatis Carolinae. Mathematica et Physica 033.2 (1992), 35-40.
%e As a word, A284622 = 011010001011010010..., in which 1 is in positions 2,3,5,9,11,...
%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 9] (* A010060 *)
%t w = StringJoin[Map[ToString, s]]
%t w1 = StringReplace[w, {"0011" -> "0"}]
%t st = ToCharacterCode[w1] - 48 (* A284622 *)
%t Flatten[Position[st, 0]] (* A284623 *)
%t Flatten[Position[st, 1]] (* A284626 *)
%Y Cf. A010060, A284622, A284623.
%K nonn,easy
%O 1,1
%A _Clark Kimberling_, May 10 2017
%E Name corrected by _Michel Dekking_, Aug 26 2017
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