0,1

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 (deutsch(AT)duke.poly.edu), Jan 10 2003

Characteristic function of A000069 (odious numbers). a(n) = 1-A010059(n) = A001285(n)-1. - Ralf Stephan (ralf(AT)ark.in-berlin.de), 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 (ctibor.zizka(AT)seznam.cz), 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 (ctibor.zizka(AT)seznam.cz), Feb 25 2008

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. Berstel and J. Karhumaki, Combinatorics on words - a tutorial, Bull. EATCS, #79 (2003), pp. 178-228.

F. Dejean, Sur un theoreme de Thue. J. Combinatorial Theory Ser. A 13 (1972), 90-99.

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.

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.

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.

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.

B. Kitchens, Review of "Computational Ergodic Theory" by G. H. Choe, Bull. Amer. Math. Soc., 44 (2007), 147-155.

Clifford A. Pickover, A Passion for Mathematics, Wiley, 2005; see p. 60.

N. J. A. Sloane, Table of n, a(n) for n = 0..16383

Joerg Arndt, Fxtbook

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

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Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (2).

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics (3).

Index entries for "core" sequences

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 (benoit7848c(AT)orange.fr), 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 (benoit7848c(AT)orange.fr), Dec 10 2003

a(n)=-1+sum(k=0, n, binomial(n, k){mod 2}) {mod 3}=-1+A001316(n) {mod 3} - Benoit Cloitre (benoit7848c(AT)orange.fr), 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 (benoit7848c(AT)orange.fr), Apr 26 2005

a(n) = A001969(n) - 2n. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Aug 28 2006

a(n) = A115384(n) - A115384(n-1) for n>0. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Aug 26 2007

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.

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 (deutsch(AT)duke.poly.edu), Mar 19 2005

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)

(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)

Cf. A001285 (for 1,2 version), A010059 (1,0 version), A048707. A010060(n)=A000120(n) mod 2.

Cf. A080813, A036581, A108694. See also the Thue (or Roth) constant A014578.

Backward first differences give A029883.

Cf. A001969.

Cf. A132680.

Cf. A115384.

Adjacent sequences: A010057 A010058 A010059 this_sequence A010061 A010062 A010063

Sequence in context: A057215 A029691 A053866 this_sequence A118247 A122257 A129950

nonn,core,easy,nice

njas

Additional comments from Robert G. Wilson v (rgwv(AT)rgwv.com), Dec 29 2000