login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A286746 {00->null}-transform of the infinite Fibonacci word A003849. 1
0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET

1

COMMENTS

As a word, A003849 = 01001010010010100101001001010010..., and deleting each occurrence of 00 gives 01101110110111011101101110110111011..., in which, it is conjectured, the positions of 0 are given by A214971, and of 1, by A195121.

Is this A230603 with an offset changed by 2? - R. J. Mathar, May 25 2017

From Michel Dekking, Aug 16 2019: (Start)

Proof of the two conjectures by Kimberling: first note that the [00->null]-transform is the same as the [001->1]-transform, since 000 does not occur in the infinite Fibonacci word.

Next, we do the following trick: replace the [001->1]-transform by the [001->2]-transform. Then the Fibonacci word is mapped to b:= A284749 = 0120122012... Note that the positions of 0 in b are the same as the positions of 0 in a. By Theorem 31 in the Allouche-Dekking paper, the positions of 0 in b are given by the sequence with terms floor(n*phi)+2*n+1, for n=0,1,2.... Transforming to offset 1, this is the sequence (ceiling((n-1)*phi) + 2*(n-1)), conjectured by Baruchel for A214971, and proved in my paper on base-phi representations.

The positions of 1 in (a(n)) are given by A195121 for n > 0, since by the Comments in A195121 this sequence has terms 3*n - floor(n*phi) = floor((3-phi)*n), and one easily verifies that phi+2 and 3-phi form a Beatty pair.

(End)

Proof of the conjecture by Mathar: this follows directly from Lemma 9.1.3 in the book by Allouche and Shallit. - Michel Dekking, Aug 16 2019

REFERENCES

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003.

LINKS

Clark Kimberling, Table of n, a(n) for n = 1..10000

J.-P. Allouche, F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.

M. Dekking, Base phi representations and golden mean beta-expansions, arXiv:1906.08437 [math.NT], 2019.

FORMULA

a(n) = floor(n/(3-phi)) - floor((n-1)/(3-phi)). - Michel Dekking, Aug 16 2019

MATHEMATICA

s = Nest[Flatten[# /. {0 -> {0, 1}, 1 -> {0}}] &, {0}, 12]; (* A003849 *)

w = StringJoin[Map[ToString, s]];

w1 = StringReplace[w, {"00" -> ""}]; st = ToCharacterCode[w1] - 48; (* A286746 *)

Flatten[Position[st, 0]];  (* A214971 *)

Flatten[Position[st, 1]];  (* A195121 *)

CROSSREFS

Cf. A003849, A214971, A195121.

Sequence in context: A123506 A051105 A284929 * A155897 A144610 A188068

Adjacent sequences:  A286743 A286744 A286745 * A286747 A286748 A286749

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, May 14 2017

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified October 4 01:04 EDT 2022. Contains 357237 sequences. (Running on oeis4.)