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 A285345 Fixed point of the morphism 0 -> 10, 1 -> 1100. 6
 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1 COMMENTS This is a 3-automatic sequence. See Allouche et al. link. - Michel Dekking, Oct 05 2020 LINKS Clark Kimberling, Table of n, a(n) for n = 1..10000 J.-P. Allouche, F. M. Dekking, and M. Queffélec, Hidden automatic sequences, arXiv:2010.00920 [math.NT], 2020. Index entries for sequences that are fixed points of mappings FORMULA Conjecture: a(n) = A284905(n+1). - R. J. Mathar, May 08 2017 From Michel Dekking, Jan 26 2024: (Start) Proof of Mathar's conjecture: let alpha be the morphism 0 -> 10, 1 -> 1100, and let beta be the morphism 0 -> 01, 1 -> 1001, which has A284905 as its fixed point starting with 0. Note that alpha^n(0) tends to (a(n)) as n tends to infinity because alpha(0) starts with 1. It therefore suffices to prove the relation (A) : 0 alpha^n(0) = beta^n(0) 0 for all n=1,2,3,... To prove such a thing one uses the fact that alpha and beta are conjugate morphisms, i.e., there exists a word u such that (B) beta(w) = u^{-1} alpha(w) u. Here u^{-1} is the free group inverse of u. We have in this case u:=1, and it suffices to prove (B) for the words w=0, and w=1. Indeed: beta(0) = 01 = 1^{-1} 10 1 = 1^{-1} alpha(0) 1, beta(1) = 1001 = 1^{-1} 1100 1 = 1^{-1} alpha(1) 1. Next, we prove (A). For n=1, we do have 0 alpha(0) = 010 = beta(0) 0. Suppose (A) has been proved till n. Then 0 alpha^{n+1}(0) = 1^{-1} 10 alpha^{n+1}(0) (10)^{-1} 10 = 1^{-1} alpha(0) alpha^{n+1}(0) (alpha(0))^{-1} 10 = 1^{-1} alpha(0 alpha^n(0) 0^{-1} ) 10 = 1^{-1} alpha(beta^n(0) ) 1 0 = beta(beta^n(0)) 0 = beta^{n+1}(0) 0. Here we used (B) with w = beta^n(0) in line three, and the induction hypothesis in line four. (End) EXAMPLE 0 -> 10-> 1100 -> 110011001010 -> ... MATHEMATICA s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {1, 1, 0, 0}}] &, {0}, 10]; (* this sequence *) Flatten[Position[s, 0]]; (* A285346 *) Flatten[Position[s, 1]]; (* A285347 *) CROSSREFS Cf. A284346, A285347. Sequence in context: A280933 A133081 A352573 * A125999 A073784 A320006 Adjacent sequences: A285342 A285343 A285344 * A285346 A285347 A285348 KEYWORD nonn,easy AUTHOR Clark Kimberling, Apr 25 2017 STATUS approved

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Last modified July 22 09:18 EDT 2024. Contains 374485 sequences. (Running on oeis4.)