A064353 Kolakoski-(1,3) sequence: the alphabet is {1,3}, and a(n) is the length of the n-th run.

1, 3, 3, 3, 1, 1, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 1, 1, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 1, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 1, 1, 1, 3, 3, 3, 1, 3, 3, 3, 1, 1, 1, 3, 3, 3, 1, 3, 3, 3, 1, 1, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 1, 1, 1, 3, 3, 3, 1, 3, 3, 3, 1, 3, 3, 3, 1, 1, 1, 3, 3, 3, 1, 3, 1, 3, 3, 3, 1, 1, 1, 3, 3

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

Historical note: the sequence (a(n)) was introduced (by me) in 1981 in a seminar in Bordeaux. It was remarked there that (a(n+1)) is a morphic sequence, i.e., a letter-to-letter projection of a fixed point of a morphism. The morphism is 1->3, 2->2, 3->343, 4->212. The letter-to-letter map is 1->1, 2->1, 3->3, 4->3. There it was also remarked that this allows one to compute the frequency of the letter 3, and an exact expression for this frequency involving sqrt(177) was given. - Michel Dekking, Jan 06 2018

The frequency of the number '3' is 0.6027847... See UWC link. - Jaap Spies, Dec 12 2004

The terms 13, 13331, 13331113331 are primes. - Vincenzo Librandi, Mar 02 2016

Consider the Kolakoski sequence generalized to the alphabet {A,B}, where A=2p+1, B=2q+1. The fraction of symbols that are A approaches f_A, calculated as follows: x=(p+q+1)/3; y=((p-q)^2)/2; lambda = x + (x^3+y+sqrt(y^2+2*x^3*y))^(1/3) + (x^3+y-sqrt(y^2+2*x^3*y))^(1/3); f_A=(lambda-2q-1)/(2p-2q). The technique is the "simple computation" mentioned by Dekking and repeated in the UWC link. - Ed Wynn, Jul 29 2019

References

• E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.
• F. M. Dekking: "What is the long range order in the Kolakoski sequence?" in: The Mathematics of Long-Range Aperiodic Order, ed. R. V. Moody, Kluwer, Dordrecht (1997), pp. 115-125.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Michael Baake and Bernd Sing, Kolakoski-(3,1) is a (deformed) model set, arXiv:math/0206098 [math.MG], 2002-2003.
• William Cook, A Recursive Block Pillar Structure in the Kolakoski Sequence K(1,3), arXiv:2504.13433 [math.CO], 2025.
• F. M. Dekking, On the structure of self-generating sequences, Seminar on Number Theory, 1980-1981 (Talence, 1980-1981), Exp. No. 31, 6 pp., Univ. Bordeaux I, Talence, 1981. Math. Rev. 83e:10075.
• F. M. Dekking, What Is the Long Range Order in the Kolakoski Sequence?, Report 95-100, Technische Universiteit Delft, 1995.
• Ulrich Reitebuch, Henriette-Sophie Lipschütz, and Konrad Polthier, Visualizing the Kolakoski Sequence, Bridges Conf. Proc.; Math., Art, Music, Architecture, Culture (2023) 481-484.
• Jaap Spies, A Bit of Math, The Art of Problem Solving, Jaap Spies Publishers (2019).
• UWC, Opgave A (solution)

Crossrefs

Cf. A000002, A071820, A071907, A071928, A071942.

See also A362927, A362928 (subtract 2 from the terms). For indices of 1's and 3's, see A362929, A362930.

Programs

Haskell

-- from John Tromp's a000002.hs
a064353 n = a064353_list !! (n-1)
a064353_list = 1 : 3 : drop 2
   (concat . zipWith replicate a064353_list . cycle $ [1, 3])
-- Reinhard Zumkeller, Aug 02 2013

MATLAB

A = [1 3 3 3]; i = 3; next = 1; while length(A) < 140 A = [A next*ones(1, A(i))]; i = i + 1; next = 4 - next; end

Mathematica

A = {1, 3, 3, 3}; i = 3; next = 1; While[Length[A] < 140, A = Join[A, next*Array[1&, A[[i]]]]; i++; next = 4-next]; A (* Jean-François Alcover, Nov 12 2016, translated from MATLAB *)

Extensions

More terms from David Wasserman, Jul 16 2002

Edited by Charles R Greathouse IV, Apr 20 2010

Restored the original definition, following a suggestion from Jianing Song. - N. J. A. Sloane, May 13 2021