A285373 Fixed point of the morphism 0 -> 10, 1 -> 1110.

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

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: A076213 A368907 A120525 * A359543 A361121 A371087

Adjacent sequences: A285370 A285371 A285372 * A285374 A285375 A285376

Keyword

nonn,easy

Author

Clark Kimberling, Apr 25 2017

Status

approved