diff --git a/thue-morse.v b/thue-morse.v index a79cd29..542a822 100644 --- a/thue-morse.v +++ b/thue-morse.v @@ -1,684 +1,7 @@ (* - -A010060 Thue-Morse sequence: let A_k denote the first 2^k terms; then A_0 = 0 and for k >= 0, A_{k+1} 541 - = A_k B_k, where B_k is obtained from A_k by interchanging 0's and 1's. - 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, - 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, - 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1 - (list; graph; refs; listen; history; text; internal format) - OFFSET 0,1 - COMMENTS Named after Axel Thue, whose name is pronounced as if it were spelled "Tü" where the ü sound - is roughly as in the German word üben. (It is incorrect to say "Too-ee" or "Too-eh".) - N. J. - A. Sloane, Jun 12 2018 - - Also called the Thue-Morse infinite word, or the Morse-Hedlund sequence, or the parity - sequence. - - Fixed point of the morphism 0 --> 01, 1 --> 10, see example. - Joerg Arndt, Mar 12 2013 - - The sequence is cubefree (does not contain three consecutive identical blocks) [see Offner for - a direct proof] and is overlap-free (does not contain XYXYX where X is 0 or 1 and Y is any - string of 0's and 1's). - - a(n) = "parity sequence" = parity of number of 1's in binary representation of n. - - To construct the sequence: alternate blocks of 0's and 1's of successive lengths A003159(k) - - A003159(k-1), k = 1, 2, 3, ... (A003159(0) = 0). Example: since the first seven differences of - A003159 are 1, 2, 1, 1, 2, 2, 2, the sequence starts with 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0. - - Emeric Deutsch, Jan 10 2003 - - Characteristic function of A000069 (odious numbers). - Ralf Stephan, Jun 20 2003 - - a(n) = S2(n) mod 2, where S2(n) = sum of digits of n, n in base-2 notation. There is a class - of generalized Thue-Morse sequences: Let Sk(n) = sum of digits of n; n in base-k notation. Let - F(t) be some arithmetic function. Then a(n)= F(Sk(n)) mod m is a generalized Thue-Morse - sequence. The classical Thue-Morse sequence is the case k=2, m=2, F(t)= 1*t. - Ctibor O. - Zizka, Feb 12 2008 (with correction from Daniel Hug, May 19 2017) - - More generally, the partial sums of the generalized Thue-Morse sequences a(n) = F(Sk(n)) mod m - are fractal, where Sk(n) is sum of digits of n, n in base k; F(t) is an arithmetic function; m - integer. - Ctibor O. Zizka, Feb 25 2008 - - Starting with offset 1, = running sums mod 2 of the kneading sequence (A035263, 1, 0, 1, 1, 1, - 0, 1, 0, 1, 0, 1, 1, 1, ...); also parity of A005187: (1, 3, 4, 7, 8, 10, 11, 15, 16, 18, 19, - ...). - Gary W. Adamson, Jun 15 2008 - - Generalized Thue-Morse sequences mod n (n>1) = the array shown in A141803. As n -> infinity - the sequences -> (1, 2, 3, ...). - Gary W. Adamson, Jul 10 2008 - - The Thue-Morse sequence for N = 3 = A053838, (sum of digits of n in base 3, mod 3): (0, 1, 2, - 1, 2, 0, 2, 0, 1, 1, 2, ...) = A004128 mod 3. - Gary W. Adamson, Aug 24 2008 - - For all positive integers k, the subsequence a(0) to a(2^k-1) is identical to the subsequence - a(2^k+2^(k-1)) to a(2^(k+1)+2^(k-1)-1). That is to say, the first half of A_k is identical to - the second half of B_k, and the second half of A_k is identical to the first quarter of - B_{k+1}, which consists of the k/2 terms immediately following B_k. - - Proof: The subsequence a(2^k+2^(k-1)) to a(2^(k+1)-1), the second half of B_k, is by - definition formed from the subsequence a(2^(k-1)) to a(2^k-1), the second half of A_k, by - interchanging its 0's and 1's. In turn, the subsequence a(2^(k-1)) to a(2^k-1), the second - half of A_k, which is by definition also B_{k-1}, is by definition formed from the subsequence - a(0) to a(2^(k-1)-1), the first half of A_k, which is by definition also A_{k-1}, by - interchanging its 0's and 1's. Interchanging the 0's and 1's of a subsequence twice leaves it - unchanged, so the subsequence a(2^k+2^(k-1)) to a(2^(k+1)-1), the second half of B_k, must be - identical to the subsequence a(0) to a(2^(k-1)-1), the first half of A_k. - - Also, the subsequence a(2^(k+1)) to a(2^(k+1)+2^(k-1)-1), the first quarter of B_{k+1}, is by - definition formed from the subsequence a(0) to a(2^(k-1)-1), the first quarter of A_{k+1}, by - interchanging its 0's and 1's. As noted above, the subsequence a(2^(k-1)) to a(2^k-1), the - second half of A_k, which is by definition also B_{k-1}, is by definition formed from the - subsequence a(0) to a(2^(k-1)-1), which is by definition A_{k-1}, by interchanging its 0's and - 1's, as well. If two subsequences are formed from the same subsequence by interchanging its - 0's and 1's then they must be identical, so the subsequence a(2^(k+1)) to - a(2^(k+1)+2^(k-1)-1), the first quarter of B_{k+1}, must be identical to the subsequence - a(2^(k-1)) to a(2^k-1), the second half of A_k. - - Therefore the subsequence a(0), ..., a(2^(k-1)-1), a(2^(k-1)), ..., a(2^k-1) is identical to - the subsequence a(2^k+2^(k-1)), ..., a(2^(k+1)-1), a(2^(k+1)), ..., a(2^(k+1)+2^(k-1)-1), QED. - - According to the German chess rules of 1929 a game of chess was drawn if the same sequence of - moves was repeated three times consecutively. Euwe, see the references, proved that this rule - could lead to infinite games. For his proof he reinvented the Thue-Morse sequence. - Johannes - W. Meijer, Feb 04 2010 - - "Thue-Morse 0->01 & 1->10, at each stage append the previous with its complement. Start with - 0, 1, 2, 3 and write them in binary. Next calculate the sum of the digits (mod 2) - that is - divide the sum by 2 and use the remainder." Pickover, The Math Book. - - Let s_2(n) be the sum of the base-2 digits of n and epsilon(n) = (-1)^s_2(n), the Thue-Morse - sequence, then prod(n >= 0, ((2*n+1)/(2*n+2))^epsilon(n) ) = 1/sqrt(2). - Jonathan Vos Post, - Jun 06 2012 - - Dekking shows that the constant obtained by interpreting this sequence as a binary expansion - is transcendental; see also "The Ubiquitous Prouhet-Thue-Morse Sequence". - Charles R - Greathouse IV, Jul 23 2013 - - Drmota, Mauduit, and Rivat proved that the subsequence a(n^2) is normal--see A228039. - - Jonathan Sondow, Sep 03 2013 - - Although the probability of a 0 or 1 is equal, guesses predicated on the latest bit seen - produce a correct match 2 out of 3 times. - Bill McEachen, Mar 13 2015 - - From a(0) to a(2n+1), there are n+1 terms equal to 0 and n+1 terms equal to 1 (see Hassan - Tarfaoui link, Concours Général 1990). - Bernard Schott, Jan 21 2022 - REFERENCES J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 15. - - Jason Bell, Michael Coons, and Eric Rowland, "The Rational-Transcendental Dichotomy of Mahler - Functions", Journal of Integer Sequences, Vol. 16 (2013), #13.2.10. - - J. Berstel and J. Karhumaki, Combinatorics on words - a tutorial, Bull. EATCS, #79 (2003), pp. - 178-228. - - B. Bollobas, The Art of Mathematics: Coffee Time in Memphis, Cambridge, 2006, p. 224. - - S. Brlek, Enumeration of factors in the Thue-Morse word, Discrete Applied Math., 24 (1989), - 83-96. doi:10.1016/0166-218X(92)90274-E. - - Yann Bugeaud and Guo-Niu Han, A combinatorial proof of the non-vanishing of Hankel - determinants of the Thue-Morse sequence, Electronic Journal of Combinatorics 21(3) (2014), - #P3.26. - - Y. Bugeaud and M. Queffélec, On Rational Approximation of the Binary Thue-Morse-Mahler Number, - Journal of Integer Sequences, 16 (2013), #13.2.3. - - J. Cooper and A. Dutle, Greedy Galois Games, Amer. Math. Mnthly, 120 (2013), 441-451. - - Currie, James D. "Non-repetitive words: Ages and essences." Combinatorica 16.1 (1996): 19-40 - - A. de Luca and S. Varricchio, Some combinatorial properties of the Thue-Morse sequence and a - problem in semigroups, Theoret. Comput. Sci. 63 (1989), 333-348. - - Colin Defant, Anti-Power Prefixes of the Thue-Morse Word Journal of Combinatorics, 24(1) - (2017), #P1.32 - - F. M. Dekking, Transcendance du nombre de Thue-Morse, Comptes Rendus de l'Academie des - Sciences de Paris 285 (1977), pp. 157-160. - - F. M. Dekking, On repetitions of blocks in binary sequences. J. Combinatorial Theory Ser. A 20 - (1976), no. 3, pp. 292-299. MR0429728(55 #2739) - - Dekking, Michel, Michel Mendès France, and Alf van der Poorten. "Folds." The Mathematical - Intelligencer, 4.3 (1982): 130-138 & front cover, and 4:4 (1982): 173-181 (printed in two - parts). - - Dubickas, Artūras. On a sequence related to that of Thue-Morse and its applications. Discrete - Math. 307 (2007), no. 9-10, 1082--1093. MR2292537 (2008b:11086). - - Fabien Durand, Julien Leroy, and Gwenaël Richomme, "Do the Properties of an S-adic - Representation Determine Factor Complexity?", Journal of Integer Sequences, Vol. 16 (2013), - #13.2.6. - - M. Euwe, Mengentheoretische Betrachtungen Über das Schachspiel, Proceedings Koninklijke - Nederlandse Akademie van Wetenschappen, Amsterdam, Vol. 32 (5): 633-642, 1929. - - S. Ferenczi, Complexity of sequences and dynamical systems, Discrete Math., 206 (1999), - 145-154. - - S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 6.8. - - W. H. Gottschalk and G. A. Hedlund, Topological Dynamics. American Mathematical Society, - Colloquium Publications, Vol. 36, Providence, RI, 1955, p. 105. - - J. Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), - 4419-4429. - - A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schroedinger - operators, Commun. Math. Phys. 174 (1995), 149-159. - - Mari Huova and Juhani Karhumäki, "On Unavoidability of k-abelian Squares in Pure Morphic - Words", Journal of Integer Sequences, Vol. 16 (2013), #13.2.9. - - B. Kitchens, Review of "Computational Ergodic Theory" by G. H. Choe, Bull. Amer. Math. Soc., - 44 (2007), 147-155. - - Le Breton, Xavier, Linear independence of automatic formal power series. Discrete Math. 306 - (2006), no. 15, 1776-1780. - - M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 23. - - Donald MacMurray, A mathematician gives an hour to chess, Chess Review 6 (No. 10, 1938), 238. - [Discusses Marston's 1938 article] - - Mauduit, Christian. Multiplicative properties of the Thue-Morse sequence. Period. Math. - Hungar. 43 (2001), no. 1-2, 137--153. MR1830572 (2002i:11081) - - Mignosi, F.; Restivo, A.; Sciortino, M. Words and forbidden factors. WORDS (Rouen, 1999). - Theoret. Comput. Sci. 273 (2002), no. 1-2, 99--117. MR1872445 (2002m:68096) - - Marston Morse, Title?, Bull. Amer. Math. Soc., 44 (No. 9, 1938), p. 632. [Mentions an - application to chess] - - C. A. Pickover, Wonders of Numbers, Adventures in Mathematics, Mind and Meaning, Chapter 17, - 'The Pipes of Papua,' Oxford University Press, Oxford, England, 2000, pages 34-38. - - C. A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60. - - Clifford A. Pickover, The Math Book, From Pythagoras to the 57th Dimension, 250 Milestones in - the History of Mathematics, Sterling Publ., NY, 2009, page 316. - - Narad Rampersad and Elise Vaslet, "On Highly Repetitive and Power Free Words", Journal of - Integer Sequences, Vol. 16 (2013), #13.2.7. - - G. Richomme, K. Saari, L. Q. Zamboni, Abelian complexity in minimal subshifts, J. London Math. - Soc. 83(1) (2011) 79-95. - - Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions - this sequence - see "List of Sequences" in Vol. 2. - - M. Rigo, P. Salimov, and E. Vandomme, "Some Properties of Abelian Return Words", Journal of - Integer Sequences, Vol. 16 (2013), #13.2.5. - - Benoit Rittaud, Elise Janvresse, Emmanuel Lesigne and Jean-Christophe Novelli, Quand les maths - se font discrètes, Le Pommier, 2008 (ISBN 978-2-7465-0370-0). - - A. Salomaa, Jewels of Formal Language Theory. Computer Science Press, Rockville, MD, 1981, p. - 6. - - Shallit, J. O. "On Infinite Products Associated with Sums of Digits." J. Number Th. 21, - 128-134, 1985. - - Ian Stewart, "Feedback", Mathematical Recreations Column, Scientific American, 274 (No. 3, - 1996), page 109 [Historical notes on this sequence] - - Thomas Stoll, On digital blocks of polynomial values and extractions in the Rudin-Shapiro - sequence, RAIRO - Theoretical Informatics and Applications (RAIRO: ITA), EDP Sciences, 2016, - 50, pp. 93-99. . - - A. Thue. Über unendliche Zeichenreihen, Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. Christiania, - No. 7 (1906), 1-22. - - A. Thue, Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen, Norske Vid. Selsk. - Skr. I. Mat. Nat. Kl. Christiania, 1 (1912), 1-67. - - S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 890. - LINKS N. J. A. Sloane, Table of n, a(n) for n = 0..16383 - - A. G. M. Ahmed, AA Weaving. In: Proceedings of Bridges 2013: Mathematics, Music, Art, ..., - 2013. - - A. Aksenov, The Newman phenomenon and Lucas sequence, arXiv preprint arXiv:1108.5352 - [math.NT], 2011-2012. - - Max A. Alekseyev and N. J. A. Sloane, On Kaprekar's Junction Numbers, arXiv:2112.14365, 2021; - Journal of Combinatorics and Number Theory, 2022 (to appear). - - J.-P. Allouche, Series and infinite products related to binary expansions of integers, - Behaviour, 4.4 (1992): p. 5. - - J.-P. Allouche, Lecture notes on automatic sequences, Krakow October 2013. - - J.-P. Allouche, Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse - sequence, J. de Théorie des Nombres de Bordeaux, 27, no. 2 (2015), 375-388. - - J.-P. Allouche, Andre Arnold, Jean Berstel, Srecko Brlek, William Jockusch, Simon Plouffe and - Bruce E. Sagan, A relative of the Thue-Morse sequence, Discrete Math., 139 (1995), 455-461. - - Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit and Luca Q. Zamboni, A Taxonomy of - Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017. - - J.-P. Allouche and H. Cohen, Dirichlet Series and Curious Infinite Products, Bull. London - Math. Soc. 17, 531-538, 1985. - - J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences, in: Axel F. and Gratias - D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, - Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11. - - J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences, in: Axel F. and Gratias - D. (eds), Beyond Quasicrystals. Centre de Physique des Houches, vol 3. Springer, Berlin, - Heidelberg, pp. 293-367, 1995; DOI https://doi.org/10.1007/978-3-662-03130-8_11. [Local copy] - - J.-P. Allouche and Jeffrey Shallit, The Ubiquitous Prouhet-Thue-Morse Sequence, in C. Ding. T. - Helleseth and H. Niederreiter, eds., Sequences and Their Applications: Proceedings of SETA - '98, Springer-Verlag, 1999, pp. 1-16. - - J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II, Theoretical Computer - Science 307.1 (2003): 3-29. doi:10.1016/S0304-3975(03)00090-2 - - Jorge Almeida and Ondrej Klíma, Binary patterns in the Prouhet-Thue-Morse sequence, - arXiv:1904.07137 [math.CO], 2019. - - Joerg Arndt, Matters Computational (The Fxtbook), p. 44. - - G. N. Arzhantseva, C. H. Cashen, D. Gruber and D. Hume, Contracting geodesics in infinitely - presented graphical small cancellation groups, arXiv preprint arXiv:1602.03767 [math.GR], - 2016-2018. - - Ricardo Astudillo, On a Class of Thue-Morse Type Sequences, J. Integer Seqs., Vol. 6, 2003. - - F. Axel et al., Vibrational modes in a one dimensional "quasi-alloy": the Morse case, J. de - Physique, Colloq. C3, Supp. to No. 7, Vol. 47 (Jul 1986), pp. C3-181-C3-186; see Eq. (10). - - M. Baake, U. Grimm and J. Nilsson, Scaling of the Thue-Morse diffraction measure, arXiv - preprint arXiv:1311.4371 [math-ph], 2013. - - Scott Balchin and Dan Rust, Computations for Symbolic Substitutions, Journal of Integer - Sequences, Vol. 20 (2017), Article 17.4.1. - - Lucilla Baldini and Josef Eschgfäller, Random functions from coupled dynamical systems, arXiv - preprint arXiv:1609.01750 [math.CO], 2016. See Example 3.11. - - J. Berstel, Axel Thue's papers on repetitions in words: a translation, July 21 1994. - Publications du LaCIM, Département de mathématiques et d'informatique 20, Université du Québec - à Montréal, 1995, 85 pages. [Cached copy] - - J.-F. Bertazzon, Resolution of an integral equation with the Thue-Morse sequence, - arXiv:1201.2502v1 [math.CO], Jan 12, 2012. - - Françoise Dejean, Sur un Théorème de Thue, J. Combinatorial Theory, vol. 13 A, iss. 1 (1972) - 90-99. - - F. Michel Dekking, Morphisms, Symbolic sequences, and their Standard Forms, arXiv:1509.00260 - [math.CO], 2015. - - E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, - J. Num. Theory 117 (2006), 191-215. - - M. Drmota, C. Mauduit and J. Rivat, The Thue-Morse Sequence Along The Squares is Normal, - Abstract, ÖMG-DMV Congress, 2013. - - Arthur Dolgopolov, Equitable Sequencing and Allocation Under Uncertainty, Preprint, 2016. - - J. Endrullis, D. Hendriks and J. W. Klop, Degrees of streams, Journal of Integers B 11 (2011): - 1-40.. - - A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of - Combinatorial Number Theory, Vol. 4, Paper G6, 2004. - - Hao Fu and G.-N. Han, Computer assisted proof for Apwenian sequences related to Hankel - determinants, arXiv preprint arXiv:1601.04370 [math.NT], 2016. - - Maciej Gawro and Maciej Ulas, "On formal inverse of the Prouhet-Thue-Morse sequence." Discrete - Mathematics 339.5 (2016): 1459-1470. Also arXiv:1601.04840, 2016. - - Michael Gilleland, Some Self-Similar Integer Sequences - - Daniel Goc, Luke Schaeffer and Jeffrey Shallit, The Subword Complexity of k-Automatic - Sequences is k-Synchronized, arXiv 1206.5352 [cs.FL], Jun 28 2012. - - G. A. Hedlund, Remarks on the work of Axel Thue on sequences, Nordisk Mat. Tid., 15 (1967), - 148-150. - - Dimitri Hendriks, Frits G. W. Dannenberg, Jorg Endrullis, Mark Dow and Jan Willem Klop, - Arithmetic Self-Similarity of Infinite Sequences, arXiv preprint 1201.3786 [math.CO], 2012. - - A. M. Hinz, S. Klavžar, U. Milutinović and C. Petr, The Tower of Hanoi - Myths and Maths, - Birkhäuser 2013. See page 79. Website for book - - Tanya Khovanova, There are no coincidences, arXiv preprint 1410.2193 [math.CO], 2014. - - Clark Kimberling, Intriguing infinite words composed of zeros and ones, Elemente der - Mathematik (2021). - - Naoki Kobayashi, Kazutaka Matsuda and Ayumi Shinohara, Functional Programs as Compressed Data, - Higher-Order and Symbolic Computation, 25, no. 1 (2012): 39-84.. - - Philip Lafrance, Narad Rampersad and Randy Yee, Some properties of a Rudin-Shapiro-like - sequence, arXiv:1408.2277 [math.CO], 2014. - - C. Mauduit, J. Rivat, La somme des chiffres des carres, Acta Mathem. 203 (1) (2009) 107-148. - - M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc., 22 - (1921), 84-100. - - K. Nakano, Shall We Juggle, Coinductively?, in Certified Programs and Proofs, Lecture Notes in - Computer Science Volume 7679, 2012, pp. 160-172. - - Hieu D. Nguyen, A mixing of Prouhet-Thue-Morse sequences and Rademacher functions, arXiv - preprint arXiv:1405.6958 [math.NT], 2014. - - Hieu D. Nguyen, A Generalization of the Digital Binomial Theorem , J. Int. Seq. 18 (2015) # - 15.5.7. - - C. D. Offner, Repetitions of Words and the Thue-Morse sequence. Preprint, no date. - - Matt Parker, The Fairest Sharing Sequence Ever, YouTube video, Nov 27 2015 - - A. Parreau, M. Rigo, E. Rowland and E. Vandomme, A new approach to the 2-regularity of the - l-abelian complexity of 2-automatic sequences, arXiv preprint arXiv:1405.3532 [cs.FL], 2014. - - C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," - Zentralblatt review - - E. Prouhet, Mémoire sur quelques relations entre les puissances des nombres, Comptes Rendus - Acad. des Sciences, 33 (No. 8, 1851), p. 225. [Said to implicitly mention this sequence] - - Michel Rigo, Relations on words, arXiv preprint arXiv:1602.03364 [cs.FL], 2016. - - Luke Schaeffer and Jeffrey Shallit, Closed, Palindromic, Rich, Privileged, Trapezoidal, and - Balanced Words in Automatic Sequences, Electronic Journal of Combinatorics 23(1) (2016), - #P1.25. - - M. R. Schroeder & N. J. A. Sloane, Correspondence, 1991 - - R. Schroeppel and R. W. Gosper, HACKMEM #122 (1972). - - Vladimir Shevelev, Two analogs of Thue-Morse sequence, arXiv preprint arXiv:1603.04434 - [math.NT], 2016-2017. - - N. J. A. Sloane, The first 1000 terms as a string - - L. Spiegelhofer, Normality of the Thue-Morse Sequence along Piatetski-Shapiro sequences, - Quart. J. Math. 66 (3) (2015). - - Hassan Tarfaoui, Concours Général 1990 - Exercice 1 (in French). - - Eric Weisstein's World of Mathematics, Thue-Morse Sequence - - Eric Weisstein's World of Mathematics, Thue-Morse Constant - - Eric Weisstein's World of Mathematics, Parity - - Joost Winter, Marcello M. Bonsangue, and Jan J. M. M. Rutten, Context-free coalgebras, Journal - of Computer and System Sciences, 81.5 (2015): 911-939. - - Hans Zantema, Complexity of Automatic Sequences, International Conference on Language and - Automata Theory and Applications (LATA 2020): Language and Automata Theory and Applications, - 260-271. - - Index entries for sequences that are fixed points of mappings - - Index entries for "core" sequences - - Index entries for sequences related to binary expansion of n - - Index entries for characteristic functions - - Index to sequences related to Olympiads and other Mathematical competitions. - FORMULA a(2n) = a(n), a(2n+1) = 1 - a(n), a(0) = 0. Also, a(k+2^m) = 1 - a(k) if 0 <= k < 2^m. - - If n = Sum b_i*2^i is the binary expansion of n then a(n) = Sum b_i (mod 2). - - Let S(0) = 0 and for k >= 1, construct S(k) from S(k-1) by mapping 0 -> 01 and 1 -> 10; - sequence is S(infinity). - - G.f.: (1/(1 - x) - Product_{k >= 0} (1 - x^(2^k)))/2. - Benoit Cloitre, Apr 23 2003 - - a(0) = 0, a(n) = (n + a(floor(n/2))) mod 2; also a(0) = 0, a(n) = (n - a(floor(n/2))) mod 2. - - Benoit Cloitre, Dec 10 2003 - - a(n) = -1 + (Sum_{k=0..n} binomial(n,k) mod 2) mod 3 = -1 + A001316(n) mod 3. - Benoit - Cloitre, May 09 2004 - - Let b(1) = 1 and b(n) = b(ceiling(n/2)) - b(floor(n/2)) then a(n-1) = (1/2)*(1 - b(2n-1)). - - Benoit Cloitre, Apr 26 2005 - - a(n) = 1 - A010059(n) = A001285(n) - 1. - Ralf Stephan, Jun 20 2003 - - a(n) = A001969(n) - 2n. - Franklin T. Adams-Watters, Aug 28 2006 - - a(n) = A115384(n) - A115384(n-1) for n > 0. - Reinhard Zumkeller, Aug 26 2007 - - For n >= 0, a(A004760(n+1)) = 1 - a(n). - Vladimir Shevelev, Apr 25 2009 - - a(A160217(n)) = 1 - a(n). - Vladimir Shevelev, May 05 2009 - - a(n) == A000069(n) (mod 2). - Robert G. Wilson v, Jan 18 2012 - - a(n) = A000035(A000120(n)). - Omar E. Pol, Oct 26 2013 - - a(n) = A000035(A193231(n)). - Antti Karttunen, Dec 27 2013 - - a(n) + A181155(n-1) = 2n for n >= 1. - Clark Kimberling, Oct 06 2014 - - G.f. A(x) satisfies: A(x) = x / (1 - x^2) + (1 - x) * A(x^2). - Ilya Gutkovskiy, Jul 29 2021 - - From Bernard Schott, Jan 21 2022: (Start) - - a(n) = a(n*2^k) for k >= 0. - - a((2^m-1)^2) = (1-(-1)^m))/2 (see Hassan Tarfaoui link, Concours Général 1990). (End) - EXAMPLE The evolution starting at 0 is: - - .0 - - .0, 1 - - .0, 1, 1, 0 - - .0, 1, 1, 0, 1, 0, 0, 1 - - .0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0 - - .0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, - 1 - - ....... - - A_2 = 0 1 1 0, so B_2 = 1 0 0 1 and A_3 = A_2 B_2 = 0 1 1 0 1 0 0 1. - - From Joerg Arndt, Mar 12 2013: (Start) - - The first steps of the iterated substitution are - - Start: 0 - - Rules: - - 0 --> 01 - - 1 --> 10 - - ------------- - - 0: (#=1) - - 0 - - 1: (#=2) - - 01 - - 2: (#=4) - - 0110 - - 3: (#=8) - - 01101001 - - 4: (#=16) - - 0110100110010110 - - 5: (#=32) - - 01101001100101101001011001101001 - - 6: (#=64) - - 0110100110010110100101100110100110010110011010010110100110010110 - - (End) - - From Omar E. Pol, Oct 28 2013: (Start) - - Written as an irregular triangle in which row lengths is A011782, the sequence begins: - - 0; - - 1; - - 1,0; - - 1,0,0,1; - - 1,0,0,1,0,1,1,0; - - 1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1; - - 1,0,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,1,0; - - It appears that: row j lists the first A011782(j) terms of A010059, with j >= 0; row sums give - A166444 which is also 0 together with A011782; right border gives A000035. - - (End) - MAPLE s := proc(k) local i, ans; ans := [ 0, 1 ]; for i from 0 to k do ans := [ op(ans), - op(map(n->(n+1) mod 2, ans)) ] od; return ans; end; t1 := s(6); A010060 := n->t1[n]; # s(k) - gives first 2^(k+2) terms. - - a := proc(k) b := [0]: for n from 1 to k do b := subs({0=[0, 1], 1=[1, 0]}, b) od: b; end; # - a(k), after the removal of the brackets, gives the first 2^k terms. # Example: a(3); gives - [[[[0, 1], [1, 0]], [[1, 0], [0, 1]]]] - - A010060:=proc(n) - - add(i, i=convert(n, base, 2)) mod 2 ; - - end proc: - - seq(A010060(n), n=0..104); # Emeric Deutsch, Mar 19 2005 - - map(`-`, convert(StringTools[ThueMorse](1000), bytes), 48); # Robert Israel, Sep 22 2014 - MATHEMATICA Table[ If[ OddQ[ Count[ IntegerDigits[n, 2], 1]], 1, 0], {n, 0, 100}]; - - mt = 0; Do[ mt = ToString[mt] <> ToString[(10^(2^n) - 1)/9 - ToExpression[mt] ], {n, 0, 6} ]; - Prepend[ RealDigits[ N[ ToExpression[mt], 2^7] ] [ [1] ], 0] - - Mod[ Count[ #, 1 ]& /@Table[ IntegerDigits[ i, 2 ], {i, 0, 2^7 - 1} ], 2 ] (* Harlan J. - Brothers, Feb 05 2005 *) - - Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {0}, 7] (* Robert G. Wilson v Sep 26 2006 - *) - - a[n_] := If[n == 0, 0, If[Mod[n, 2] == 0, a[n/2], 1 - a[(n - 1)/2]]] (* Ben Branman, Oct 22 - 2010 *) - - a[n_] := Mod[Length[FixedPointList[BitAnd[#, # - 1] &, n]], 2] (* Jan Mangaldan, Jul 23 2015 - *) - - Table[2/3 (1 - Cos[Pi/3 (n - Sum[(-1)^Binomial[n, k], {k, 1, n}])]), {n, 0, 100}] (* or, for - version 10.2 or higher *) Table[ThueMorse[n], {n, 0, 100}] (* Vladimir Reshetnikov, May 06 - 2016 *) - - ThueMorse[Range[0, 100]] (* The program uses the ThueMorse function from Mathematica version - 11 *) (* Harvey P. Dale, Aug 11 2016 *) - PROG (Haskell) - - a010060 n = a010060_list !! n - - a010060_list = - - 0 : interleave (complement a010060_list) (tail a010060_list) - - where complement = map (1 - ) - - interleave (x:xs) ys = x : interleave ys xs - - -- Doug McIlroy (doug(AT)cs.dartmouth.edu), Jun 29 2003 - - -- Edited by Reinhard Zumkeller, Oct 03 2012 - - (PARI) a(n)=if(n<1, 0, sum(k=0, length(binary(n))-1, bittest(n, k))%2) - - (PARI) a(n)=if(n<1, 0, subst(Pol(binary(n)), x, 1)%2) - - (PARI) default(realprecision, 6100); x=0.0; m=20080; for (n=1, m-1, x=x+x; x=x+sum(k=0, - length(binary(n))-1, bittest(n, k))%2); x=2*x/2^m; for (n=0, 20000, d=floor(x); x=(x-d)*2; - write("b010060.txt", n, " ", d)); \\ Harry J. Smith, Apr 28 2009 - - (PARI) a(n)=hammingweight(n)%2 \\ Charles R Greathouse IV, Mar 22 2013 - - (Python) - - A010060_list = [0] - - for _ in range(14): - - A010060_list += [1-d for d in A010060_list] # Chai Wah Wu, Mar 04 2016 - - (R) - - maxrow <- 8 # by choice - - b01 <- 1 - - for(m in 0:maxrow) for(k in 0:(2^m-1)){ - - b01[2^(m+1)+ k] <- b01[2^m+k] - - b01[2^(m+1)+2^m+k] <- 1-b01[2^m+k] - - } - - (b01 <- c(0, b01)) - - # Yosu Yurramendi, Apr 10 2017 - CROSSREFS Cf. A001285 (for 1, 2 version), A010059 (for 1, 0 version), A106400 (for +1, -1 version), - A048707. A010060(n)=A000120(n) mod 2. - - Cf. A007413, A080813, A080814, A036581, A108694. See also the Thue (or Roth) constant A014578, - also A014571. - - Cf. also A001969, A035263, A005187, A115384, A132680, A141803, A104248, A193231. - - Run lengths give A026465. Backward first differences give A029883. - - Cf. A004128, A053838, A059448, A171900, A161916, A214212, A005942 (subword complexity), - A010693 (Abelian complexity), A225186 (squares), A228039 (a(n^2)), A282317. - - Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: - A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, - 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first - term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, - 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: - A316829, 37: A010060. - - Sequence in context: A143222 A286490 A217831 * A316569 A284848 A286484 - - Adjacent sequences: A010057 A010058 A010059 * A010061 A010062 A010063 - KEYWORD nonn,core,easy,nice - AUTHOR N. J. A. Sloane - STATUS approved - - Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam - Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents - The OEIS Community | Maintained by The OEIS Foundation Inc. - - License Agreements, Terms of Use, Privacy Policy. . - - Last modified November 20 14:06 EST 2022. Contains 358247 sequences. (Running on oeis4.) - - *) +https://oeis.org/A010060 +https://en.wikipedia.org/wiki/Thue%E2%80%93Morse_sequence +*) Require Import Coq.Lists.List. Require Import PeanoNat. @@ -1152,24 +475,17 @@ Proof. + rewrite Nat.mul_lt_mono_pos_r with (p := 2^m) in H1. rewrite <- Nat.pow_add_r in H1. rewrite Nat.add_comm. assumption. rewrite <- Nat.neq_0_lt_0. apply Nat.pow_nonzero. easy. - + rewrite nth_error_app1. rewrite nth_error_app1. + + assert (L: forall l, l < 2^m -> k*2^m + l < length (tm_step (m+n))). + intro l. intro M. rewrite tm_size_power2. + destruct k. simpl. assert (2^m <= 2^(m+n)). + apply Nat.pow_le_mono_r. easy. apply Nat.le_add_r. + generalize H3. generalize M. apply Nat.lt_le_trans. + apply lt_split_bits. apply Nat.lt_0_succ. assumption. + assumption. + + rewrite nth_error_app1. rewrite nth_error_app1. generalize H1. apply IHn. - - rewrite tm_size_power2. - assert (k * 2^m + j < 2^(m+n)). - destruct k. simpl. assert (2^m <= 2^(m+n)). - apply Nat.pow_le_mono_r. easy. apply Nat.le_add_r. - generalize H3. generalize I. apply Nat.lt_le_trans. - apply lt_split_bits. apply Nat.lt_0_succ. - assumption. assumption. assumption. - - rewrite tm_size_power2. - assert (k * 2^m + i < 2^(m+n)). - destruct k. simpl. assert (2^m <= 2^(m+n)). - apply Nat.pow_le_mono_r. easy. apply Nat.le_add_r. - generalize H3. generalize H. apply Nat.lt_le_trans. - apply lt_split_bits. apply Nat.lt_0_succ. - assumption. assumption. assumption. + apply L. assumption. apply L. assumption. + assert (J: 2 ^ (m + n) <= k * 2 ^ m). rewrite Nat.pow_add_r. rewrite Nat.mul_comm. @@ -1308,6 +624,16 @@ Proof. + + + + + + + + (* TODO: supprimer head_2 *) + + Require Import BinNat.