

A064353


Kolakoski(1,3) sequence: a(n) is length of nth run.


12



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
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OFFSET

1,2


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 lettertoletter projection of a fixed point of a morphism. The morphism is 1>3, 2>2, 3>343, 4>212. The lettertoletter map is 1>1, 2>1, 3>3, 4>3. Here it was also remarked that this admits 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
13, 13331, 13331113331 are primes.  Vincenzo Librandi, Mar 02 2016


REFERENCES

F. M. Dekking: "What is the long range order in the Kolakoski sequence?" in: The Mathematics of LongRange Aperiodic Order, ed. R. V. Moody, Kluwer, Dordrecht (1997), pp. 115125.
E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 3235, Volume 59 (Jeux math'), April/June 2008, Paris.


LINKS

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
F. M. Dekking, On the structure of selfgenerating sequences, Seminar on Number Theory, 19801981 (Talence, 19801981), 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 95100, Technische Universiteit Delft, 1995.
Michael Baake and Bernd Sing, Kolakoski(3,1) is a (deformed) model set, arXiv:math/0206098 [math.MG], 20022003.
UWC, Opgave A (solution)


MATHEMATICA

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


PROG

(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
(Haskell)  from John Tromp's a000002.hs
a064353 n = a064353_list !! (n1)
a064353_list = 1 : 3 : drop 2
(concat . zipWith replicate a064353_list . cycle $ [1, 3])
 Reinhard Zumkeller, Aug 02 2013


CROSSREFS

Cf. A000002, A071820, A071907, A071928, A071942.
Sequence in context: A177693 A131289 A130974 * A190906 A080311 A135368
Adjacent sequences: A064350 A064351 A064352 * A064354 A064355 A064356


KEYWORD

nonn,easy,nice


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from David Wasserman, Jul 16 2002
Edited by Charles R Greathouse IV, Apr 20 2010


STATUS

approved



