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A Necessary Condition for the Existence of the Asymptotic Density in Kolakoski Sequence

Abdallah Hammam

Département de Mathématiques, Faculté des sciences, Université Moulay Ismaïl, 50020 Meknès, Morocco

Abstract: We investigate here the Oldenburger-Kolakoski sequence with K₁=1. In the first part, we give some expressions of the discrepancy function δ(n) representing the difference between 2s and 1s in K₁K₂…Kn. The discrepancy could be interpreted as a perturbation of a certain equilibrium. Our main result is a necessary and sufficient condition for the existence of the asymptotic density. In the last section, we present an algorithm to generate the sequence terms and a formula for the term Kn.

Keywords: Kolakoski sequence, recursive formula, asymptotic density

Received January 08, 2018; Revised March 03, 2018; Accepted April 24, 2018

Citation: Abdallah Hammam, A Necessary Condition for the Existence of the Asymptotic Density in Kolakoski Sequence. Turkish Journal of Analysis and Number Theory, 6 (2). 57-60. January 2018. doi:10.12691/tjant-6-2-5

1. Introduction

The Kolakoski sequence with defines the unique infinite word which does not change by the Run Length Encoding operator It starts like this: 122112122122112... Many questions asked by Kimberling [2] about this sequence, are still open. For example, it is conjectured by Keane [1] that asymptotically, there is as many one’s as two’s, which means that the limiting density of 2s is Other information in concern with this sequence can be found in the Sloane’s OEIS [3].

2. Notation

We now introduce some definitions and notation.

We let and denote respectively the run-length encoding operator, defined by the following example, and its inverse. The Kolakoski infinite word can be seen at the same time as a fixed point of and of for The density of 2s in is given by

We define the successive partial sums by

and

We will use the following cardinals and are respectively the numbers of 1s and 2s in and with having respectively the same parity than

For instance,

= is the number of 2s in with even indices.

is the number of 1s in with even index and odd partial sum.

is the number of 2s in with even index j, odd partial sum and even

3. Some Expressions of the Discrepancy

The discrepancy function represents the difference between the number of 2s and the number of 1s in the word It can be expressed as follows

Proposition 1.

Proof. If we replace by in the expression of use the fact that

we get

From here, we conclude that if the limiting density exists and is not then the discrepancies and should have the same sign for large enough n.

4. The Discrepancy is a Perturbation of an Equilibrium

In this section, we will consider the integers n such that are even

Proposition 2.

Figure 1. The sign of the discrepancy changes

Proposition 3.

Proposition 4.

The proof of these propositions is based on the identity where we replace by and

The discrepancy function can be seen as perturbation of some equilibrium. Hence, and are three perturbations of three equilibriums and there is no reason they will have same sign. The change of the sign of the discrepancy means that if the density has a limit, it will be

5. A Necessary Condition for the Existence of the Asymptotic Density

Let and be the respective densities of and in

Proposition 5.

In particular,

Proof. Using the definition of the 2s density, we have

but

thus

If we assume that the asymptotic density exist and , then, as a subsequence, we should have and consequently

On the other hand

By the same, we will have

which gives the condition

6. An Interesting Algorithm to Generate the Sequence Terms

A natural way to generate the sequence is the one based on a first integration using the inverse run-length encoding operator The blue ones create ones while the red ones create twos. The blue twos will generate blocs of two ones and the red twos will give blocs of two twos.

To make the algorithm faster, we will use the third integration defined by This is illustrated as follows

For the multi-run generated by after three integrations, will depend on the parity of and Its length will be

We can improve the speed of the algorithm with the integration but the computations become complex.

So, with if we know we can generate where

7. A New Formula for the nth Term Kn

Lemma 6. For every

Proof. By definition, we have

replacing n by we get and Using the expression of given above, the difference yields to

If we replace again n by the first identity becomes

which yields to

and finally gives

Proposition 7. Let and p the unique integer such that

If we assume known the terms then, using the recursive formulas above, we can compute and and predict the word generated by represented by the number after three integrations. The difference gives the position and the following value of

Where is a number which depends on the parity of and as illustrated by the table below.

Table 1. The number Np generated by Kp,p,SP and Δ₁⁻³

8. Concluding Remarks

The question about the existence of the limiting density is still unanswered. We just replaced it by an equivalent one concerning the density of even twos. The expressions of the discrepancy presented here show the complexity of Kolakoski sequence and justify why the answer to Keane conjecture is not easy. We believe that is not possible to have an explicit expression of the term All one can do, is to find from some previous terms with as small as possible. We proved that improving the former result In the end, the advantage of the algorithm used to generate is that, it could be easily generalized and made faster and faster.

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. I also thank some of my friends for fruitful discussions.

References

[1]   | Keane and C. Series (eds.), Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford University Press, 1991, pp. 35-70.
[2]   | C.Kimberling,http://faculty.evansville.edu/ck6/integer/index.html.
[3]   | N. J. A. Sloane, The On-line Encyclopedia of Integer Sequences, published electronically at http://oeis.org.