The Oldenburger-Kolakoski Sequence

Statement:

The Oldenburger-Kolakoski Sequence, A000002 is given by \(a(n)\) is the length of the n-th run; \(a(1)=1\). $$122112122122112112212112122\ldots$$ Question: If a string \(x_1x_2 \ldots x_n\) occurs, must its reverse \(x_n \ldots x_2x_1\) occur later in the sequence?

Author	Date Published	Prize	Paid By
Oldenburger-Kolakoski	1939	$200.00	Clark Kimberling [1]

Partial Results:

Related Problems:



References:

[1] Clark Kimberling Unsolved Problems and Rewards

[2] J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 337.

[3] E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.

[4] F. M. Dekking, What Is the Long Range Order in the Kolakoski Sequence?, in The mathematics of long-range aperiodic order (Waterloo, ON, 1995), 115-125, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 489, Kluwer Acad. Publ., Dordrecht, 1997. Math. Rev. 98g:11022.

[5] M. S. Keane, Ergodic theory and subshifts of finite type, Chap. 2 of T. Bedford et al., eds., Ergodic Theory, Symbolic Dynamics and Hyperbolic Spaces, Oxford, 1991, esp. p. 50.

[6] J. C. Lagarias, Number Theory and Dynamical Systems, pp. 35-72 of S. A. Burr, ed., The Unreasonable Effectiveness of Number Theory, Proc. Sympos. Appl. Math., 46 (1992). Amer. Math. Soc.

[7] Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.

[8] N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

[9] N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

[10] I. Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 233.

Keywords:

OEIS; Oldenburger-Kolakoski Sequence; string;