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A064353
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Kolakoski-(1,3) sequence: the alphabet is {1,3}, and a(n) is the length of the n-th run.
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16
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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
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1,2
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COMMENTS
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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
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
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REFERENCES
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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.
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LINKS
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Ulrich Reitebuch, Henriette-Sophie Lipschütz, and Konrad Polthier, Visualizing the Kolakoski Sequence, Bridges Conf. Proc.; Math., Art, Music, Architecture, Culture (2023) 481-484.
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MATHEMATICA
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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 *)
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PROG
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(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 !! (n-1)
a064353_list = 1 : 3 : drop 2
(concat . zipWith replicate a064353_list . cycle $ [1, 3])
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CROSSREFS
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KEYWORD
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nonn,easy,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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