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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

Index entries for sequences that are fixed points of mappings

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.)