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A096268 Period-doubling sequence (or period-doubling word): fixed point of the morphism 0 -> 01, 1 -> 00. 37
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.
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 first 200000 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
Equivalently, with offset 1, the characteristic function of A036554 and an indicator for the A003159/A036554 classification of positive integers. - Peter Munn, Jun 02 2020
REFERENCES
Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
LINKS
N. J. A. Sloane, Table of n, a(n) for n = 0..10000 (first 1022 terms from T. D. Noe)
J.-P. Allouche, M. Baake, J. Cassaigns, and D. Damanik, Palindrome complexity, arXiv:math/0106121 [math.CO], 2001; Theoretical Computer Science, 292 (2003), 9-31.
Scott Balchin and Dan Rust, Computations for Symbolic Substitutions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.1.
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 and C. Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 79. Book's website
Laszlo Mérai and A. Winterhof, On the Nth linear complexity of automatic sequences, arXiv preprint arXiv:1711.10764 [math.NT], 2017.
Jeong-Yup Lee, D. Flom and S. I. Ben-Abraham, Multidimensional period doubling structures, Acta Crystallographica Section A: Foundations, (2016). A72, 391-394.
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:1405.3532 [cs.FL], 2014-2015.
Narad Rampersad and Manon Stipulanti, The Formal Inverse of the Period-Doubling Sequence, arXiv:1807.11899 [math.CO], 2018.
Luke Schaeffer and Jeffrey Shallit, Closed, Palindromic, Rich, Privileged, Trapezoidal, and Balanced Words in Automatic Sequences, Electronic Journal of Combinatorics 23(1) (2016), Article P1.25.
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
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1/3. = Amiram Eldar, Sep 18 2022
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
# second Maple program:
a:= proc(n) a(n):= `if`(n::even, 0, 1-a((n-1)/2)) end:
seq(a(n), n=0..125); # Alois P. Heinz, Mar 20 2019
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
(Python)
def A096268(n): return (~(n+1)&n).bit_length()&1 # Chai Wah Wu, Jan 09 2023
CROSSREFS
Not the same as A073059!
Swapping 0 and 1 gives A035263.
Cf. A056832, A123087 (partial sums).
With offset 1, classification indicator for A003159/A036554.
Also with offset 1: A007814 mod 2 (cf. A096271 for mod 3), A048675 mod 2 (cf. A332813 for mod 3), A059975 mod 2.
Sequence in context: A285969 A189664 A260446 * A219097 A284751 A288426
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 February 27 10:59 EST 2024. Contains 370378 sequences. (Running on oeis4.)