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A096268 Period-doubling sequence (or period-doubling word): fixed point of the morphism 0 -> 01, 1 -> 00. 23
0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,1

COMMENTS

Take highest power of 2 dividing n (A007814(n+1)), read modulo 2.

For the scale-invariance properties see Hendriks et al., 2012.

From Paolo P. Lava , Apr 14 2008: (Start)

At the m-th step (m=0,1,2,3,..., starting with 0 at step m=0) form the concatenation of the partial sequence (of length 2^m) with itself changing only the last digit (1 -> 0, 0 ->1). Thus

m=0 -> 0

m=1 -> 0 U 1 -> 01

m=2 -> 01 U 00 -> 0100

m=3 -> 0100 U 0101 -> 01000101

m=4 -> 01000101 U 01000100 -> 0100010101000100

etc. (End)

This is the sequence that results from the ternary Thue-Morse sequence (A036577) if all twos in that sequence are replaced by zeros. - Nathan Fox, Mar 12 2013

This sequence can be used to draw the Von Koch snowflake with a suitable walk in the plane. Start from the origin then the n-th step is "turn +Pi/3 if a(n)=0 and turn -2*Pi/3 if a(n)=1" (see link for a plot of the 200000 first steps). - Benoit Cloitre, Nov 10 2013

1 iff the number of trailing zeros in the binary representation of n+1 is odd. - Ralf Stephan, Nov 11 2013

REFERENCES

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

LINKS

T. D. Noe, Table of n, a(n) for n=0..1022

J.-P. Allouche, M. Baake, J. Cassaigns and D. Damanik, Palindrome complexity, arXiv:math/0106121 [math.CO], 2001.

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

Benoit Cloitre, 200000 steps in the plane using "turn +Pi/3 if a(n)=0 and -2Pi/3 otherwise"

F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.

G. J. Endrullis, D. Hendriks and J. W. Klop, Degrees of streams

Dimitri Hendriks, Frits G. W. Dannenberg, Jorg Endrullis, Mark Dow and Jan Willem Klop, Arithmetic Self-Similarity of Infinite Sequences, arXiv 1201.3786 [math.CO], 2012

A. M. Hinz, S. Klavžar, U. Milutinović, C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 79. Book's website

Jeong-Yup Lee, D Flom, SI Ben-Abraham, Multidimensional period doubling structures, Acta Crystallographica Section A: Foundations, (2016). A72, 391-394.

A. Parreau, M. Rigo, E. Rowland, E. Vandomme, A new approach to the 2-regularity of the l-abelian complexity of 2-automatic sequences, arXiv:1405.3532 [cs.FL], 2014-2015.

Luke Schaeffer, Jeffrey Shallit, Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences, Electronic Journal of Combinatorics 23(1) (2016), #P1.25.

Index entries for sequences that are fixed points of mappings

Index entries for sequences related to binary expansion of n

FORMULA

Recurrence: a(2*n) = 0, a(4*n+1) = 1, a(4*n+3) = a(n). - Ralf Stephan, Dec 11 2004

The recurrence may be extended backwards, with a(-1) = 1. - S. I. Ben-Abraham, Apr 01 2013

a(n) = 1 - A035263(n-1). - Reinhard Zumkeller, Aug 16 2006

Dirichlet g.f.: zeta(s)/(1+2^s). - Ralf Stephan, Jun 17 2007

Let T(x) be the g.f., then T(x)+T(x^2)=x^2/(1-x^2). - Joerg Arndt, May 11 2010

Let 2^k||n+1. Then a(n)=1 if k is odd, a(n)=0 if k is even. - Vladimir Shevelev, Aug 25 2010

a(n) = A007814(n+1)(mod 2). - Robert G. Wilson v, Jan 18 2012

a((2*n+1)*2^p-1) = p mod 2, p >= 0 and n >= 0. - Johannes W. Meijer, Feb 02 2013

a(n) = A056832(n+1) - 1. - Reinhard Zumkeller, Jul 29 2014

EXAMPLE

Start: 0

Rules:

  0 --> 01

  1 --> 00

-------------

0:   (#=1)

  0

1:   (#=2)

  01

2:   (#=4)

  0100

3:   (#=8)

  01000101

4:   (#=16)

  0100010101000100

5:   (#=32)

  01000101010001000100010101000101

6:   (#=64)

  0100010101000100010001010100010101000101010001000100010101000100

7:   (#=128)

  010001010100010001000101010001010100010101000100010001010100010001000101010...

[Joerg Arndt, Jul 06 2011]

MAPLE

nmax:=104: for p from 0 to ceil(simplify(log[2](nmax))) do for n from 0 to ceil(nmax/(p+2))+1 do a((2*n+1)*2^p-1) := p mod 2 od: od: seq(a(n), n=0..nmax); # Johannes W. Meijer, Feb 02 2013

MATHEMATICA

Nest[ Flatten[ # /. {0 -> {1, 0}, 1 -> {0, 0}}] &, {1}, 7] (* Robert G. Wilson v, Mar 05 2005 *)

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

PROG

(PARI) a(n)=valuation(n+1, 2)%2 \\ Ralf Stephan, Nov 11 2013

(Haskell)

a096268 = (subtract 1) . a056832 . (+ 1)

-- Reinhard Zumkeller, Jul 29 2014

(MAGMA) [Valuation(n+1, 2) mod 2: n in [0..100]]; // Vincenzo Librandi, Jul 20 2016

CROSSREFS

Not the same as A073059!

Swapping 0 and 1 gives A035263.

Cf. A096269, A096270, A071858, A096271, A220466, A036577.

Cf. A056832, A123087 (partial sums).

Sequence in context: A110161 A134667 A117943 * A079101 A076478 A091444

Adjacent sequences:  A096265 A096266 A096267 * A096269 A096270 A096271

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Jun 22 2004

EXTENSIONS

Corrected by Jeremy Gardiner, Dec 12 2004

More terms from Robert G. Wilson v, Feb 26 2005

STATUS

approved

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Last modified January 22 22:38 EST 2018. Contains 298093 sequences.