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 A285373 Fixed point of the morphism 0 -> 10, 1 -> 1110. 5
 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1 COMMENTS From  Michel Dekking, Jan 22 2018: (Start) Proof of Mathar's conjecture of May 08 2017: Let sigma be the morphism    sigma:  0 -> 10, 1 -> 1110, and let tau be the morphism      tau:  0 -> 01, 1 -> 1101, which has A284939 as a fixed point. It clearly suffices to prove the relation       (A) : 0 sigma^n(0) = tau^n(0) 0   for all n=1,2,... To prove such a thing one needs a second relation, for example,       (B) : sigma^n(10) = sigma^n(0) 0^{-1} tau^n(1) 0  for all n=1,2,... Here 0^{-1} is the free group inverse of 0. Note that (A) and (B) together imply       (C) :  sigma^n(10) = 0^{-1} tau^n(01) 0   for all n=1,2,... But, since sigma(0)=10, and tau(0)=01, relation (C) is equal to relation (A), with n replaced by n+1. It is therefore enough to prove (B) by induction. Well, (A), (B) and (C) are easily checked for n=1. Furthermore, using the induction hypothesis with (B) and (C) in the first line, and again (C) in the third line, one obtains sigma^{n+1}(10)   = sigma^n(11)sigma^n(01)sigma^n(01)   = sigma^n(1)sigma^n(1) sigma^n(0) 0^{-1} tau^n(1) 0 0^{-1} tau^n(01) 0   = sigma^n(1)sigma^n(10) 0^{-1} tau^n(101) 0   = sigma^n(1) sigma^n(0) 0^{-1} tau^n(1) 0 0^{-1} tau^n(101) 0   = sigma^n(10) 0^{-1} tau^n(1101) 0   = sigma^{n+1}(0) 0^{-1} tau^{n+1}(1) 0. (End) LINKS Clark Kimberling, Table of n, a(n) for n = 1..10000 FORMULA Conjecture: a(n) = A284939(n+1). - R. J. Mathar, May 08 2017 EXAMPLE 0 -> 10-> 1110 -> 11101110111010 -> MATHEMATICA s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {1, 1, 1, 0}}] &, {0}, 10] (* A285373 *) Flatten[Position[s, 0]]  (* A285374 *) Flatten[Position[s, 1]]  (* A285375 *) CROSSREFS Cf. A284374, A285375. Sequence in context: A132350 A076213 A120525 * A112299 A230901 A285671 Adjacent sequences:  A285370 A285371 A285372 * A285374 A285375 A285376 KEYWORD nonn,easy AUTHOR Clark Kimberling, Apr 25 2017 STATUS approved

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Last modified November 17 13:12 EST 2019. Contains 329230 sequences. (Running on oeis4.)