A000002 - OEIS

%I M0190 N0070 #389 Nov 16 2024 09:06:55

%S 1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2,1,1,2,1,1,2,1,

%T 2,2,1,2,2,1,1,2,1,2,2,1,2,1,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,1,1,

%U 2,1,1,2,1,2,2,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,1,2,2

%N Kolakoski sequence: a(n) is length of n-th run; a(1) = 1; sequence consists just of 1's and 2's.

%C Historical note: the sequence might be better called the Oldenburger-Kolakoski sequence, since it was discussed by Rufus Oldenburger in 1939; see links. - _Clark Kimberling_, Dec 06 2012.  However, to avoid confusion, this sequence will be known in the OEIS as the Kolakoski sequence. It is undesirable to have some entries refer to the Oldenburger-Kolakoski sequence and others to the Kolakoski sequence. - _N. J. A. Sloane_, Nov 22 2017

%C It is an unsolved problem to show that the density of 1's is equal to 1/2.

%C A weaker problem is to construct a combinatorial bijection between the set of positions of 1's and the set of positions of 2's. - _Gus Wiseman_, Mar 01 2016

%C The sequence is cubefree and all square subwords have lengths which are one of 2, 4, 6, 18 and 54 (see A294447) [Carpi, 1994].

%C This is a fractal sequence: replace each run with its length and recover the original sequence. - _Kerry Mitchell_, Dec 08 2005

%C Kupin and Rowland write: We use a method of Goulden and Jackson to bound freq_1(K), the limiting frequency of 1 in the Kolakoski word K. We prove that |freq_1(K) - 1/2| <= 17/762, assuming the limit exists and establish the semirigorous bound |freq_1(K) - 1/2| <= 1/46. - _Jonathan Vos Post_, Sep 16 2008

%C freq_1(K) is conjectured to be 1/2 + O(log(K)) (see PlanetMath link). - _Jon Perry_, Oct 29 2014

%C Conjecture: Taking the sequence in word lengths of 10, for example, batch 1-10, 11-20, etc., then there can only be 4, 5 or 6 1's in each batch. - _Jon Perry_, Sep 26 2012

%C From _Jean-Christophe Hervé_, Oct 04 2014: (Start)

%C The sequence does not contain words of the form ababa, because this would imply the impossible 111 (1 b, 1 a, 1 b) somewhere before. This demonstrates the conjecture made by _Jon Perry_: more than 6 1's or 6 2's in a word of 10 would necessitate something like aabaabaaba, which would imply the impossible 12121 before (word aabaababaa is also impossible because of ababa). The remark on the sextuplets below even shows that the number of 1's in any 9-tuplet is always 4 or 5.

%C There are only 6 triples that appear in the sequence (112, 121, 122, 211, 212 and 221); and by the preceding argument, only 18 sextuplets: the 6 double triples (112112, etc.); 112122, 112212, 121122, 121221, 211212, and 211221; and those obtained by reversing the order of the triples (122112, etc.). Regarding the density of 1's in the sequence, these 12 sextuplets all have a density 1/2 of 1's, and the 6 double triples all lead to a word with this exact density after transformation by the Kolakoski rules, for example: 112112 -> 12112122 (4 1's/8); this is because the second triple reverses the numbers of 1's and 2's generated by the first triple. Therefore, the sequence can be split into the double triples on one side, a part whose transformation (which is in the sequence) has a density of 1's of 1/2; and a part with the other sextuplets, which has directly the same density of 1's. (End)

%C If we map 1 to +1 and 2 to -1, then the mapped sequence would have a [conjectured] mean of 0, since the Kolakoski sequence is [conjectured] to have an equal density (1/2) of 1s and 2s. For the partial sums of this mapped sequence, see A088568. - _Daniel Forgues_, Jul 08 2015

%C Looking at the plot for A088568, it seems that although the asymptotic densities of 1s and 2s appear to be 1/2, there might be a bias in favor of the 2s. I.e., D(1) = 1/2 - O(log(n)/n), D(2) = 1/2 + O(log(n)/n). - _Daniel Forgues_, Jul 11 2015

%C From _Michel Dekking_, Jan 31 2018: (Start)

%C (a(n)) is the unique fixed point of the 2-block substitution beta

%C     11 -> 12

%C     12 -> 122

%C     21 -> 112

%C     22 -> 1122.

%C A 2-block substitution beta maps a word w(1)...w(2n) to the word

%C     beta(w(1)w(2))...beta(w(2n-1)w(2n)).

%C If the word has odd length, then the last letter is ignored.

%C It was noted by me in 1979 in the Bordeaux seminar on number theory that (a(n+1)) is fixed point of the 2-block substitution 11 -> 21, 12 -> 211, 21 -> 221,  22 -> 2211. (End)

%C Named after the American artist and recreational mathematician William George Kolakoski (1944-1997). - _Amiram Eldar_, Jun 17 2021

%D Jean-Paul Allouche and Jeffrey Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 337.

%D Éric Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.

%D 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.

%D Michael 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.

%D 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.

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

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

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

%D Ilan Vardi, Computational Recreations in Mathematica. Addison-Wesley, Redwood City, CA, 1991, p. 233.

%H N. J. A. Sloane, <a href="/A000002/b000002.txt">Table of n, a(n) for n = 1..10502</a>

%H Jean-Paul Allouche, Michael Baake, Julien Cassaigne and David Damanik, <a href="https://arxiv.org/abs/math/0106121">Palindrome complexity</a>, arXiv:math/0106121 [math.CO], 2001; <a href="https://doi.org/10.1016/S0304-3975(01)00212-2">Theoretical Computer Science</a>, Vol. 292 (2003), pp. 9-31.

%H Michael Baake and Bernd Sing, <a href="https://arxiv.org/abs/math/0206098">Kolakoski-(3,1) is a (deformed) model set</a>, arXiv:math/0206098 [math.MG], 2002-2003.

%H Alex Bellos and Brady Haran, <a href="https://www.youtube.com/watch?v=co5sOgZ3XcM">The Kolakoski Sequence</a>, Numberphile video (2017).

%H Olivier Bordellès and Benoit Cloitre, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL14/Bordelles/bordelles7r.html">Bounds for the Kolakoski Sequence</a>, J. Integer Sequences, Vol. 14 (2011), Article 11.2.1.

%H Richard P. Brent, <a href="https://maths-people.anu.edu.au/~brent/pd/Kolakoski-UNSW.pdf">Fast algorithms for the Kolakoski sequence</a>, Slides from a talk, 2016.

%H Éric Brier, Rémi Géraud-Stewart, David Naccache, Alessandro Pacco, and Emanuele Troiani, <a href="https://arxiv.org/abs/2006.07246">The Look-and-Say The Biggest Sequence Eventually Cycles</a>, arXiv:2006.07246 [math.DS], 2020.

%H Arturo Carpi, <a href="https://doi.org/10.1016/0020-0190(94)00162-6">On repeated factors in C^infinity-words</a>, Information Processing Letters, Vol. 52 (1994), pp. 289-294.

%H Benoit Cloitre, <a href="/A000002/a000002.pdf">The Kolakoski transform of words</a>.

%H Benoit Cloitre, <a href="/A000002/a000002.png">Plot of walk on the first 60000 terms (step of unit length moving right with angle Pi/2 if 2 and left with angle -Pi/2 if 1 starting at (0,0))</a>.

%H F. M. Dekking, <a href="https://www.jstor.org/stable/44165352">Regularity and irregularity of sequences generated by automata</a>, Seminar on Number Theory, 1979-1980 (Talence, 1979-1980), Exp. No. 9, 10 pp., Univ. Bordeaux I, Talence, 1980.

%H F. M. Dekking, <a href="http://www.digizeitschriften.de/dms/resolveppn/?PPN=GDZPPN002544490">On the structure of self-generating sequences</a>, Seminar on Number Theory, 1980-1981 (Talence, 1980-1981), Exp. No. 31, 6 pp., Univ. Bordeaux I, Talence, 1981. Math. Rev. 83e:10075.

%H F. M. Dekking, <a href="http://citeseerx.ist.psu.edu/pdf/7af40df61b38208d1eccca350f4869b6f1a6a18f">What Is the Long Range Order in the Kolakoski Sequence?</a>, Report 95-100, Technische Universiteit Delft, 1995.

%H F. M. Dekking, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Dekking/dek25.html">The Thue-Morse Sequence in Base 3/2</a>, J. Int. Seq., Vol. 26 (2023), Article 23.2.3.

%H Jörg Endrullis, Dimitri Hendriks and Jan Willem Klop, <a href="http://math.colgate.edu/~integers/a6num/a6num.pdf">Degrees of Streams</a>, Integers, Vol. 11B (2011), A6.

%H David Eppstein, <a href="https://11011110.github.io/blog/2016/10/13/kolakoski-tree.html">The Kolakoski tree</a> and <a href="https://11011110.github.io/blog/2016/10/14/kolakoski-sequence-via.html">The Kolakoski sequence via bit tricks instead of recursion</a>.

%H Jean-Marc Fédou and Gabriele Fici, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Fici/fici.html">Some remarks on differentiable sequences and recursivity</a>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.3.2.

%H Abdallah Hammam, <a href="http://pubs.sciepub.com/tjant/4/3/1/">Some new Formulas for the Kolakoski Sequence A000002</a>, Turkish Journal of Analysis and Number Theory, Vol. 4, No. 3 (2016), pp. 54-59.

%H Mari Huova and Juhani Karhumäki, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL16/Huova/huova2.html">On Unavoidability of k-abelian Squares in Pure Morphic Words</a>, Journal of Integer Sequences, Vol. 16 (2013), #13.2.9.

%H Clark Kimberling, Integer Sequences and Arrays, <a href="http://faculty.evansville.edu/ck6/integer/index.html">Illustration of the Kolakoski sequence</a>.

%H William Kolakoski, <a href="https://www.jstor.org/stable/2313883">Problem 5304</a>, Amer. Math. Monthly, Vol. 72, No. 8 (1965), p. 674; <a href="https://www.jstor.org/stable/2314839">Self Generating Runs</a>, Solution to Problem 5304 by Necdet Üçoluk, Vol. 73, No. 6 (1966), pp. 681-682.

%H Leonid V. Kovalev, <a href="https://web.archive.org/web/20130606153459 /http://calculus7.org/2012/04/14/kolakoski-sequence-ii/">Kolakoski sequence II</a>.

%H Elizabeth J. Kupin and Eric S. Rowland, <a href="https://arxiv.org/abs/0809.2776">Bounds on the frequency of 1 in the Kolakoski word</a>, arXiv:0809.2776 [math.CO], Sep 16, 2008.

%H Rabie A. Mahmoud, <a href="https://www.researchgate.net/publication/271910838">Hardware Implementation of Binary Kolakoski Sequence</a>, Research Gate, 2015.

%H Kerry Mitchell, <a href="http://kerrymitchellart.com/articles/Spirolateral-Type_Images_from_Integer_Sequences.pdf">Spirolateral-Type Images from Integer Sequences</a>, 2013.

%H Johan Nilsson, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL15/Nilsson/nilsson5.html">A Space Efficient Algorithm for the Calculation of the Digit Distribution in the Kolakoski Sequence</a>, J. Int. Seq., Vol. 15 (2012), Article 12.6.7; <a href="https://arxiv.org/abs/1110.4228">arXiv preprint</a>, arXiv:1110.4228 [math.CO], Oct 19, 2011.

%H J. Nilsson, <a href="https://doi.org/10.12693/APhysPolA.126.549">Letter Frequencies in the Kolakoski Sequence</a>, Acta Physica Polonica A, Vol. 126 (2014), pp. 549-552.

%H Rufus Oldenburger, <a href="https://doi.org/10.1090/S0002-9947-1939-0000352-9">Exponent trajectories in symbolic dynamics</a>, Trans. Amer. Math. Soc., Vol. 46 (1939), pp. 453-466.

%H Matthew Parker, <a href="https://oeis.org/A000002/a000002_50K.7z">The first 50K terms (7-Zip compressed file)</a>.

%H Gheorghe Păun and Arto Salomaa, <a href="https://www.jstor.org/stable/2975113">Self-reading sequences</a>, Amer. Math. Monthly, Vol. 103, No. 2 (1996), pp. 166-168.

%H Michael Rao, <a href="https://www.arthy.org/kola/kola.php">Trucs et bidules sur la séquence de Kolakoski</a>, 2012, in French.

%H A. Scolnicov, <a href="http://planetmath.org/kolakoskisequence">Kolakoski sequence</a>, PlanetMath.org.

%H Bobby Shen, <a href="https://arxiv.org/abs/1703.00180">A uniformness conjecture of the Kolakoski sequence, graph connectivity, and correlations</a>, arXiv:1703.00180 [math.CO], 2017.

%H Bernd Sing, <a href="https://emis.de/journals/INTEGERS/papers/a14num/a14num.Abstract.html">More Kolakoski Sequences</a>, INTEGERS, Vol. 11B (2011), A14.

%H N. J. A. Sloane, <a href="/A001149/a001149.pdf">Handwritten notes on Self-Generating Sequences, 1970</a>. (note that A1148 has now become A005282)

%H N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, <a href="https://vimeo.com/314786942">Part I</a>, <a href="https://vimeo.com/314790822">Part 2</a>, <a href="https://oeis.org/A320487/a320487.pdf">Slides</a>. (Mentions this sequence)

%H Bertran Steinsky, <a href="https://www.cs.uwaterloo.ca/journals/JIS/VOL9/Steinsky/steinsky5.html">A Recursive Formula for the Kolakoski Sequence A000002</a>, J. Integer Sequences, Vol. 9 (2006), Article 06.3.7.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/KolakoskiSequence.html">Kolakoski Sequence</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Kolakoski_sequence">Kolakoski sequence</a>.

%H Gus Wiseman, <a href="https://imgur.com/MqRlpnm">Kolakoski fractal animation for n=40000</a>.

%H Ed Wynn, <a href="/A000002/a000002.spl.txt">Program to generate A000002 in Shakespeare Programming Language</a>.

%H <a href="/index/Cor#core">Index entries for "core" sequences</a>

%F These two formulas define completely the sequence: a(1)=1, a(2)=2, a(a(1) + a(2) + ... + a(k)) = (3 + (-1)^k)/2 and a(a(1) + a(2) + ... + a(k) + 1) = (3 - (-1)^k)/2. - _Benoit Cloitre_, Oct 06 2003

%F a(n+2)*a(n+1)*a(n)/2 = a(n+2) + a(n+1) + a(n) - 3 (this formula doesn't define the sequence, it is just a consequence of the definition). - _Benoit Cloitre_, Nov 17 2003

%F a(n+1) = 3 - a(n) + (a(n) - a(n-1))*(a(b(n)) - 1), where b(n) is the sequence A156253. - Jean-Marc Fedou and _Gabriele Fici_, Mar 18 2010

%F a(n) = (3 + (-1)^A156253(n))/2. - _Benoit Cloitre_, Sep 17 2013

%F Conjectures from _Boštjan Gec_, Oct 07 2024: (Start)

%F a(n)*(a(n-1) + a(n-2) - 3) + a(n-1)*a(n-2) + 7 = 3*a(n-1) + 3*a(n-2).

%F a(n)*(a(n-1) + a(n-2) - 3) = a(n-3)*(a(n-1) + a(n-2) - 3). (End)

%F Comment from _Kevin Ryde_, Oct 07 2024: The above formulas are true: The parts identify when terms are same or different and they hold for any sequence of 1's and 2's with run lengths 1 or 2.

%e Start with a(1) = 1. By definition of the sequence, this says that the first run has length 1, so it must be a single 1, and a(2) = 2. Thus, the second run (which starts with this 2) must have length 2, so the third term must be also be a(3) = 2, and the fourth term can't be a 2, so must be a(4) = 1. Since a(3) = 2, the third run must have length 2, so we deduce a(5) = 1, a(6) = 2, and so on. The correction I made was to change a(4) to a(5) and a(5) to a(6). - _Labos Elemer_, corrected by _Graeme McRae_

%p M := 100; s := [ 1,2,2 ]; for n from 3 to M do for i from 1 to s[ n ] do s := [ op(s),1+((n-1)mod 2) ]; od: od: s; A000002 := n->s[n];

%p # alternative implementation based on the Cloitre formula:

%p A000002 := proc(n)

%p     local ksu,k ;

%p     option remember;

%p     if n = 1 then

%p         1;

%p     elif n <=3 then

%p         2;

%p     else

%p         for k from 1 do

%p             ksu := add(procname(i),i=1..k) ;

%p             if n = ksu then

%p                 return (3+(-1)^k)/2 ;

%p             elif n = ksu+ 1 then

%p                 return (3-(-1)^k)/2 ;

%p             end if;

%p         end do:

%p     end if;

%p end proc: # _R. J. Mathar_, Nov 15 2014

%t a[steps_] := Module[{a = {1, 2, 2}}, Do[a = Append[a, 1 + Mod[(n - 1), 2]], {n, 3, steps}, {i, a[[n]]}]; a]

%t a[ n_] := If[ n < 3, Max[ 0, n], Module[ {an = {1, 2, 2}, m = 3}, While[ Length[ an] < n, an = Join[ an, Table[ Mod[m, 2, 1], { an[[ m]]} ]]; m++]; an[[n]]]] (* _Michael Somos_, Jul 11 2011 *)

%t n=8; Prepend[ Nest[ Flatten[ Partition[#, 2] /. {{2, 2} -> {2, 2, 1, 1}, {2, 1} -> {2, 2, 1}, {1, 2} -> {2, 1, 1}, {1, 1} -> {2, 1}}] &, {2, 2}, n], 1] (* _Birkas Gyorgy_, Jul 10 2012 *)

%t KolakoskiSeq[n_Integer] := Block[{a = {1, 2, 2}}, Fold[Join[#1, ConstantArray[Mod[#2, 2] + 1, #1[[#2]]]] &, a, Range[3, n]]]; KolakoskiSeq[999] (* _Mikk Heidemaa_, Nov 01 2024 *)

%o (PARI) my(a=[1,2,2]); for(n=3,80, for(i=1,a[n],a=concat(a,2-n%2))); a

%o (PARI) {a(n) = local(an=[1, 2, 2], m=3); if( n<1, 0, while( #an < n, an = concat( an, vector(an[m], i, 2-m%2)); m++); an[n])};

%o (Haskell) a = 1:2: drop 2 (concat . zipWith replicate a . cycle $ [1,2]) -- _John Tromp_, Apr 09 2011

%o (Python)

%o # For explanation see link.

%o def Kolakoski():

%o     x = y = -1

%o     while True:

%o         yield [2,1][x&1]

%o         f = y &~ (y+1)

%o         x ^= f

%o         y = (y+1) | (f & (x>>1))

%o K = Kolakoski()

%o print([next(K) for _ in range(100)]) # _David Eppstein_, Oct 15 2016

%Y Cf. A001083, A006928, A042942, A069864, A010060, A078929, A171899, A054353 (partial sums), A074286, A216345, A294447.

%Y Cf. A054354, bisections: A100428, A100429.

%Y Cf. A013947, A156077, A234322 (positions, running total and percentage of 1's).

%Y Cf. A118270.

%Y Cf. A049705, A088569 (are either subsequences of A000002? - _Jon Perry_, Oct 30 2014)

%Y Kolakoski-type sequences using other seeds than (1,2):

%Y A078880 (2,1), A064353 (1,3), A071820 (2,3), A074804 (3,2), A071907 (1,4), A071928 (2,4), A071942 (3,4), A074803 (4,2), A079729 (1,2,3), A079730 (1,2,3,4).

%Y Other self-describing: A001462 (Golomb sequence, see also references therein), A005041, A100144.

%Y Cf. A088568 (partial sums of [3 - 2 * a(n)]).

%K nonn,core,easy,nice

%O 1,2

%A _N. J. A. Sloane_

%E Minor edits to example and PARI code made by _M. F. Hasler_, May 07 2014