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A010059 Another version of the Thue-Morse sequence: let A_k denote the first 2^k terms; then A_0 = 1 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. 45
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, 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, 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Characteristic function of A001969 (evil numbers). - Ralf Stephan, Jun 20 2003

REFERENCES

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.

M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 23.

A. Salomaa, Jewels of Formal Language Theory. Computer Science Press, Rockville, MD, 1981, p. 6.

LINKS

Robert Israel, Table of n, a(n) for n = 0..10000

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.

Scott Balchin and Dan Rust, Computations for Symbolic Substitutions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.1.

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, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.

Michael Gilleland, Some Self-Similar Integer Sequences

Tanya Khovanova, There are no coincidences, arXiv preprint 1410.2193 [math.CO], 2014.

M. Morse, Recurrent geodesics on a surface of negative curvature, Trans. Amer. Math. Soc., 22 (1921), 84-100.

Stephen Wolfram, A New Kind Of Science | Online.

Index entries for sequences related to binary expansion of n

Index entries for characteristic functions

FORMULA

G.f.: 1/2 * (1/(1-x) + prod(k>=0, 1-x^2^k)). - Ralf Stephan, Jun 20 2003

a(n) + A010060(n) = 1 for all n.

a(n) = A159481(n+1) - A159481(n). [Reinhard Zumkeller, Apr 16 2009]

a(n) + A026147(n-1) = 2n for n >= 1. - Clark Kimberling, Oct 06 2014

a(n) = A000069(n+1) (mod 2). - John M. Campbell, Jun 30 2016

a(n) = A059448(A054429(n))  = (A106400(n)+1)/2 = (1+A008836(A005940(1+n)))/2. - Antti Karttunen, May 30 2017

EXAMPLE

The evolution starting at 1 is:

.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

...........

MAPLE

A010059 := n->1-A010060(n);

map(t->49-t, convert(StringTools[ThueMorse](1000), bytes)); # Robert Israel, Feb 02 2016

MATHEMATICA

Mod[ CoefficientList[Series[(1 + Sqrt[(1 - 3x)/(1 + x)])/(2(1 + x)), {x, 0, 111}], x], 2] (* Stephan Wolfram *)

CoefficientList[ Series[1/(1 - x) + Product[1 - x^2^k, {k, 0, 10}], {x, 0, 111}]/2, x] (* Robert G. Wilson v, Jul 16 2004 *)

Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {1, 0}}] &, {1}, 7] (* Robert G. Wilson v Sep 26 2006 *)

Nest[ Join[ #, Mod[2# + 1, 3]] &, {1}, 7] (* Robert G. Wilson v, Jul 27 2014 *)

{{1}}~Join~SubstitutionSystem[{0 -> {0, 1}, 1 -> {1, 0}}, {0}, 6] // Flatten (* Michael De Vlieger, Aug 15 2016, Version 10.2 *)

PROG

(Haskell) a010059 = (1 -) . a010060  -- Reinhard Zumkeller, Feb 04 2013

(PARI) a(n)=!(hammingweight(n)%2); \\ Charles R Greathouse IV, Mar 29 2013

(R)

maxrow <- 8 # by choice

b01 <- 0

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(1, b01))

# Yosu Yurramendi, Apr 10 2017

CROSSREFS

Cf. A001285 (1, 2 version), A010060 (0, 1 version), A106400 (+1, -1 version), A059448 (with reversed subsections).

Cf. also A000069, A026147, A159481.

Sequence in context: A005171 A076404 * A143580 A011749 A188578 A104105

Adjacent sequences:  A010056 A010057 A010058 * A010060 A010061 A010062

KEYWORD

nonn,easy,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 21 15:31 EDT 2017. Contains 292310 sequences.