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A285341 Fixed point of the morphism 0 -> 10, 1 -> 1011. 5
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, 1, 0, 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

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) = A284893(n+1). - R. J. Mathar, May 08 2017

From Michel Dekking, Feb 05 2018: (Start)

Proof of this conjecture: let sigma be the morphism 0 -> 10, 1 -> 1011, and let tau be the morphism 0 -> 01, 1 -> 0111, which has A284893 as a fixed point. It clearly suffices to prove the relation, for all n=1,2,3,...:

      (A) : 0 sigma^n(0) = tau^n(0) 0

To prove such a thing one needs a second relation, for all n=1,2,3,...:

      (B) : 0 sigma^n(1) 0^{-1} = tau^n(0) tau^n(1) [tau^n(0)]^{-1}

Here 0^{-1} and [tau^n(0)]^{-1} are the free group inverses of 0 and tau^n(0).

For n=1, we do indeed have:

    (a) 0 sigma(10) = 0101110 = 0101110 = tau(01)0

    (b) 0 sigma(1) 0^{-1} tau(0) = 010111 = tau(0)tau(1).

Using the induction hypothesis with (A) and (B) in the second line, one obtains

       0 sigma^{n+1}(0) = 0 sigma^n(1) 0^{-1} 0 sigma^n(0)

                        = tau^n(0) tau^n(1) [tau^n(0)]^{-1} tau^n(0) 0

                        = tau^{n+1}(0) 0.

Similarly,

       0 sigma^{n+1}(1) 0^{-1}

        = 0 sigma^n(1) 0^{-1} 0 sigma^n(0) 0^{-1} 0 sigma^n(11) 0^{-1}

        = tau^n(0)tau^n(1)[tau^n(0)]^{-1} tau^n(0) 0 0^{-1}

       tau^n(0) tau^n(11) [tau^n(0)]^{-1}

        = tau^{n+1}(0) tau^{n+1}(1) [tau^n(1)]^{-1}[tau^n(0)]^{-1}

        = tau^{n+1}(0) tau^{n+1}(1) [tau^{n+1}(0)]^{-1}. (End)

EXAMPLE

0 -> 10-> 1011 -> 10111010111011.

MATHEMATICA

s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {1, 0, 1, 1}}] &, {0}, 10]; (* A285341 *)

u = Flatten[Position[s, 0]];  (* A285342 *)

Flatten[Position[s, 1]];  (* A285343 *)

u/2 (* A285344)

CROSSREFS

Cf. A285342, A285343, A285344.

Sequence in context: A267053 A259024 A259599 * A317198 A284677 A191232

Adjacent sequences:  A285338 A285339 A285340 * A285342 A285343 A285344

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Apr 25 2017

STATUS

approved

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Last modified September 22 11:13 EDT 2018. Contains 315270 sequences. (Running on oeis4.)