|
| |
|
|
A010060
|
|
Thue-Morse sequence: let A_k denote the first 2^k terms; then A_0 = 0 and for k >= 0, A_{k+1} = A_k B_k, where B_k is obtained from A_k by interchanging 0's and 1's.
|
|
241
| |
|
|
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; internal format)
|
|
|
|
OFFSET
| 0,1
|
|
|
COMMENTS
| Also called the Thue-Morse infinite word.
The sequence is cube-free (does not contain three consecutive identical blocks) 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). a(n) = 1-A010059(n) = A001285(n)-1. - 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 generalised Thue-Morse sequence. The classical Thue-Morse sequence is the case k=2, m=2, F(t)= 1. - Ctibor O. Zizka, Feb 12 2008
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 -> Inf. 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. [From 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 0s and 1s. 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 0s and 1s. Interchanging the 0s and 1s 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 0s and 1s. 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 0s and 1s, as well. If two subsequences are formed from the same subsequence by interchanging its 0s and 1s 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. [From 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.
|
|
|
REFERENCES
| A. Aksenov, The Newman phenomenon and Lucas sequence, Arxiv preprint arXiv:1108.5352, 2011
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.
J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 15.
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).
J.-L. Baril, Classical sequences revisited with permutations avoiding dotted pattern, Electronic Journal of Combinatorics, 18 (2011), #P178; http://www.combinatorics.org/Volume_18/PDF/v18i1p178.pdf.
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.
F. Dejean, Sur un theoreme de Thue. J. Combinatorial Theory Ser. A 13 (1972), 90-99.
M. Euwe, Mengentheoretische Betrachtungen Ueber das Schachspiel, Proceedings Koninklijke Nederlandse Akademie van Wetenschappen, Amsterdam, Vol. 32 (5): 633-642, 1929. [From Johannes W. Meijer (meijgia(AT)hotmail.com), Feb 04 2010.]
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.
G. A. Hedlund, Remarks on the work of Axel Thue on sequences, Nordisk Mat. Tid., 15 (1967), 148-150.
A. Hof, O. Knill and B. Simon, Singular continuous spectrum for palindromic Schroedinger operators, Commun. Math. Phys. 174 (1995), 149-159.
B. Kitchens, Review of "Computational Ergodic Theory" by G. H. Choe, Bull. Amer. Math. Soc., 44 (2007), 147-155.
Naoki Kobayashi, Kazutaka Matsuda and Ayumi Shinohara, Functional Programs as Compressed Data, http://www.kb.ecei.tohoku.ac.jp/~koba/papers/pepm2012-full.pdf
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.
M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc., 22 (1921), 84-100.
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.
Benoit Rittaud, Elise Janvresse, Emmanuel Lesigne and Jean-Christophe Novelli, Quand les maths se font discretes, Le Pommier, 2008 (ISBN 978-2-7465-0370-0).
A. Salomaa, Jewels of Formal Language Theory. Computer Science Press, Rockville, MD, 1981, p. 6.
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
Joerg Arndt, Fxtbook, p.44
J.-P. Allouche, Andre Arnold, Jean Berstel, Srecko Brlek, William Jockusch, Simon Plouffe and Bruce E. Sagan, A sequence related to that of Thue-Morse, Discrete Math., 139 (1995), 455-461.
J.-P. Allouche and M. Mendes France, Automata and Automatic Sequences.
J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II
J.-P. Allouche and J. O. 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.
Ricardo Astudillo, On a Class of Thue-Morse Type Sequences, J. Integer Seqs., Vol. 6, 2003.
Jean Berstel, Home Page
Bertazzon J.-F, Resolution of an integral equation with the Thue-Morse sequence, arXiv:1201.2502v1 [math.CO], Jan 12, 2012.
E. Deutsch and B. E. Sagan, Congruences for Catalan and Motzkin numbers and related sequences, J. Num. Theory 117 (2006), 191-215.
A. S. Fraenkel, Home Page
A. S. Fraenkel, New games related to old and new sequences, INTEGERS, Electronic J. of Combinatorial Number Theory, Vol. 4, Paper G6, 2004.
Michael Gilleland, Some Self-Similar Integer Sequences
M. Morse, Recurrent geodesics on a surface of negative curvature (page images), Trans. Amer. Math. Soc., 22 (1921), 84-100.
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review
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
Index entries for "core" sequences
Index entries for characteristic functions
Bertazzon J.-F, Resolution of an integral equation with the Thue-Morse sequence, arXiv:1201.2502v1 [math.CO], Jan 12, 2012.
|
|
|
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.
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(ceil(n/2))-b(floor(n/2)) then a(n-1)=(1/2)*(1-b(2n-1)) - Benoit Cloitre, Apr 26 2005
a(n) = A001969(n) - 2n. - Frank 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). [From Vladimir Shevelev, Apr 25 2009]
a(A160217(n))=1-a(n). [From Vladimir Shevelev, May 05 2009]
A010060(n) == A000069(n)(mod 2). - Robert G. Wilson v, Jan 18 2012.
|
|
|
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.
|
|
|
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]]]]
a:=proc(n) local n2: n2:=convert(n, base, 2): sum(n2[j], j=1..nops(n2)) mod 2; end: seq(a(n), n=0..104); - Emeric Deutsch, Mar 19 2005
|
|
|
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 ] (from 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]]] [From Ben Branman, Oct 22 2010]
|
|
|
PROG
| (Haskell) a = 0: interleave (complement a) (tail a) where {complement = map (1 - ); interleave (x:xs) ys = x: interleave ys xs} (from Doug McIlroy (doug(AT)cs.dartmouth.edu), Jun 29 2003)
(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)); } [From Harry J. Smith (hjsmithh(AT)sbcglobal.net), Apr 28 2009]
|
|
|
CROSSREFS
| Cf. A001285 (for 1, 2 version), A010059 (1, 0 version), A048707. A010060(n)=A000120(n) mod 2.
Cf. A007413, A080813, A080814, A036581, A108694. See also the Thue (or Roth) constant A014578.
Cf. also A001969, A035263, A005187, A115384, A132680, A141803, A104248.
Backward first differences give A029883.
Cf. A004128, A053838, A059448, A171900, A161916.
Sequence in context: A053866 A156595 A143222 * A118247 A122257 A129950
Adjacent sequences: A010057 A010058 A010059 * A010061 A010062 A010063
|
|
|
KEYWORD
| nonn,core,easy,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| Additional comments from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 29 2000
|
| |
|
|
