Abstract

The aim of our investigation is an attempt to answer two still unsolved questions about Kolakoski sequence (K_n)_n≥1: Is there an explicit expression of the n^th term K_n, and the second one, known as the conjecture of Keane, claims that the asymptotic density of twos, is In the first section of this paper, we present a new formula for K_n according to K₁, K₂, …K_p where In the second part, we define three sequences satisfying the condition U_iV_i=W_i, and using the fact that (V_i) increases at least exponentially while (W_i) does not, we conclude that (U_i) should converge to zero. Our argument is inductive but so strong to insure the validity of the conjecture in concern with density of twos.

2000 Mathematics Subject Classification: Primary 11B83; Secondary 11B85, 11Y55, 40A05.

1. Introduction

The infinite Kolakoski word as defined in Sloane's OEIS ¹, is the unique fixed point, starting by $ 1, $ of the Run Length Encoding operator :

Many questions about this self-generating sequence are still with no answer ².

The two most known are:

1. Is there an explicit expression of with respect to ?

2. Is the asymptotic density of twos,

Bordellès ³ gave expressions of and

To complete this, we add a new expression of

Steinsky ⁴ found a complicated formula which allows to compute from the former terms .

We improve this by using only with

Concerning the asymptotic density question, Steinsky ⁴ presented a curve which disapprove the conjecture, but, in our work, we use the fact that the exponential grows very fast to highly support it.

2. Notation

The successive partial sums

For and , we define the following very useful partial sums

and

For example,

The density of twos in and the discrepancy

For , by definition, and are respectively the total number of twos and the difference between the twos and the ones, in the word

For instance,

We will also need the classical identities

Remark 1. It is well known that

3. A First Expression of K_n

Lemma 2. For each integer there exists an integer such that

Proof 3. Let be a positive integer. It is clear that

So, for any integer , There are two cases:

Or

On the other hand, and

Lemma 4. For each integer ,

Proof 5. We just need results in Table 1.

Using lemmas above, one can deduce an expression of in both cases.

Corollary 6. For n ≥ 3, let

and

Then

In the next section, we give a improved expression of according to instead of

4. A Second Expression of K_n

Lemma 7. For each integer , there exists an integer such that

Proof 8. It is a simple consequence of the fact that

Corollary 9. For n ≥ 3, let

and

then

with

Proof 10.

It is easy to check that

If , then and

If then and

If then

We can replace the three cases be the next unique formula and using the fact that we get the desired expression for .

This formula has been validated by the code below, written in Maple language.

:

end do:

5. A Great Support for the Conjecture of Keane

Proposition 11. For an even natural number satisfying

Proof 12. By definition of the density ,

and

Corollary 13. From the proposition 11 just above, we see that the left hand sides contain a sequence which grows exponentially from n to while, in the right hand side, there is probable change of sign indicating no exponential increasing as illustrated in Figure 1.

6. Concluding Remarks

We presented an optimal expression of the form with , improving so some former results. About the asymptotic density of twos, We have a strong reason to support Keane's conjecture: Our argument is based on Proposition $12$. It uses the fact that if the product of two sequences have a changing sign and if increases exponentially to , then should converge to zero. This reasoning has been applied to other Kolakoski sequences: and and we obtained the results predicted by Hammam ⁵ as illustrated in Figure 2. The blue curve shows clearly that the density of twos, in goes to

7. Acknowledgments

The author would like to express his gratitude to the referees for their careful reading of the manuscript and for giving me some of their precious time.

References