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%I #26 Nov 25 2019 01:01:41
%S 0,1,1,0,1,1,1,0,1,1,0,1,1,1,0,1,1,1,0,1,1,0,1,1,1,0,1,1,0,1,1,1,0,1,
%T 1,1,0,1,1,0,1,1,1,0,1,1,1,0,1,1,0,1,1,1,0,1,1,0,1,1,1,0,1,1,1,0,1,1,
%U 0,1,1,1,0,1,1,0,1,1,1,0,1,1,1,0,1,1
%N {00->null}-transform of the infinite Fibonacci word A003849.
%C As a word, A003849 = 01001010010010100101001001010010..., and deleting each occurrence of 00 gives 01101110110111011101101110110111011..., in which, it is conjectured, the positions of 0 are given by A214971, and of 1, by A195121.
%C Is this A230603 with an offset changed by 2? - _R. J. Mathar_, May 25 2017
%C From _Michel Dekking_, Aug 16 2019: (Start)
%C Proof of the two conjectures by Kimberling: first note that the [00->null]-transform is the same as the [001->1]-transform, since 000 does not occur in the infinite Fibonacci word.
%C Next, we do the following trick: replace the [001->1]-transform by the [001->2]-transform. Then the Fibonacci word is mapped to b:= A284749 = 0120122012... Note that the positions of 0 in b are the same as the positions of 0 in a. By Theorem 31 in the Allouche-Dekking paper, the positions of 0 in b are given by the sequence with terms floor(n*phi)+2*n+1, for n=0,1,2.... Transforming to offset 1, this is the sequence (ceiling((n-1)*phi) + 2*(n-1)), conjectured by Baruchel for A214971, and proved in my paper on base-phi representations.
%C The positions of 1 in (a(n)) are given by A195121 for n > 0, since by the Comments in A195121 this sequence has terms 3*n - floor(n*phi) = floor((3-phi)*n), and one easily verifies that phi+2 and 3-phi form a Beatty pair.
%C (End)
%C Proof of the conjecture by Mathar: this follows directly from Lemma 9.1.3 in the book by Allouche and Shallit. - _Michel Dekking_, Aug 16 2019
%D J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003.
%H Clark Kimberling, <a href="/A286746/b286746.txt">Table of n, a(n) for n = 1..10000</a>
%H J.-P. Allouche, F. M. Dekking, <a href="https://arxiv.org/abs/1809.03424">Generalized Beatty sequences and complementary triples</a>, arXiv:1809.03424 [math.NT], 2018.
%H M. Dekking, <a href="https://arxiv.org/abs/1906.08437">Base phi representations and golden mean beta-expansions</a>, arXiv:1906.08437 [math.NT], 2019.
%F a(n) = floor(n/(3-phi)) - floor((n-1)/(3-phi)). - _Michel Dekking_, Aug 16 2019
%t s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 12]; (* A003849 *)
%t w = StringJoin[Map[ToString, s]];
%t w1 = StringReplace[w, {"00" -> ""}]; st = ToCharacterCode[w1] - 48; (* A286746 *)
%t Flatten[Position[st, 0]]; (* A214971 *)
%t Flatten[Position[st, 1]]; (* A195121 *)
%Y Cf. A003849, A214971, A195121.
%K nonn,easy
%O 1
%A _Clark Kimberling_, May 14 2017
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