A285301 Fixed point of the morphism 0 -> 10, 1 -> 1000.

1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0

Offset

1

Comments

Prefixing 0 gives A284751.

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) = A284751(n+1). - R. J. Mathar, May 08 2017

From Michel Dekking, Sep 11 2019: (Start)

Proof of Mathar's conjecture.

Let sigma be the morphism 0 -> 10, 1 -> 1000.

Let tau be the morphism 0 -> 01, 1 -> 0001.

Then A284751 is the fixed point of tau. So it suffices to prove that

0 sigma^n(1) = tau^n(0) 0 for all n>0.

This formula follows by induction, using that tau and sigma are conjugate morphisms: 1 tau(w) = sigma(w) 1 for all words w.

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

(End)

Example

0 -> 10-> 100010 -> 1000101010100010 ->

Mathematica

s = Nest[Flatten[# /. {0 -> {1, 0}, 1 -> {1, 0, 0, 0}}] &, {0}, 10]; (* A285301 *)
Flatten[Position[s, 0]];  (* A285302 *)
Flatten[Position[s, 1]];  (* A086398 *)

Crossrefs

Cf. A284302, A086398, A284751.

Sequence in context: A267145 A141728 A141737 * A089011 A267043 A266444

Adjacent sequences: A285298 A285299 A285300 * A285302 A285303 A285304

Keyword

nonn,easy

Author

Clark Kimberling, Apr 25 2017

Status

approved