A123087 -id:A123087 - cp's OEIS frontend

A096268 Period-doubling sequence (or period-doubling word): fixed point of the morphism 0 -> 01, 1 -> 00.

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

Offset: 0

N. J. A. Sloane, Jun 22 2004

nonn

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

			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]

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

N. J. A. Sloane, Table of n, a(n) for n = 0..10000 (first 1022 terms from T. D. Noe)
Jean-Paul Allouche, Michael Baake, Julien Cassaigne, and David 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.
Benoit Cloitre, 200000 steps in the plane using "turn +Pi/3 if a(n)=0 and -2Pi/3 otherwise".
Cristian Cobeli, Mihai Prunescu, and Alexandru Zaharescu, On non-holonomicity, transcendence and p-adic valuations, arXiv:2412.16517 [math.NT], 2024. See p. 12.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
G. Jörg Endrullis, Dimitri Hendriks, and Jan Willem Klop, Degrees of streams.
Robbert Fokkink and Gandhar Joshi, Anti-recurrence sequences, arXiv:2506.13337 [math.NT], 2025. See pp. 2, 18.
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.
Andreas M. Hinz, Sandi Klavžar, Uroš Milutinović, and Ciril Petr, The Tower of Hanoi - Myths and Maths, Birkhäuser 2013. See page 79. Book's website
Shuo Li, Palindromic length sequence of the ruler sequence and of the period-doubling sequence, arXiv:2007.08317 [math.CO], 2020.
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.
Aline Parreau, Michel Rigo, Eric Rowland, and Elise Vandomme, A new approach to the 2-regularity of the l-abelian complexity of 2-automatic sequences, arXiv:1405.3532 [cs.FL], 2014-2015.
Saúl Pilatowsky-Cameo, Soonwon Choi, and Wen Wei Ho, Critically slow Hilbert-space ergodicity in quantum morphic drives, arXiv:2502.06936 [quant-ph], 2025. See pp. 6, 15.
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.
Index entries for sequences that are fixed points of mappings.
Index entries for sequences related to binary expansion of n.

Not the same as A073059!
Swapping 0 and 1 gives A035263.
Cf. A096269, A096270, A071858, A220466, A036577.
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.

a096268 = (subtract 1) . a056832 . (+ 1)
-- Reinhard Zumkeller, Jul 29 2014

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

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

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 *)

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

def A096268(n): return (~(n+1)&n).bit_length()&1 # Chai Wah Wu, Jan 09 2023

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

Corrected by Jeremy Gardiner, Dec 12 2004

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

A050292 a(2n) = 2n - a(n), a(2n+1) = 2n + 1 - a(n) (for n >= 0).

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 9, 10, 11, 12, 12, 13, 14, 15, 15, 16, 16, 17, 17, 18, 19, 20, 20, 21, 21, 22, 22, 23, 24, 25, 25, 26, 26, 27, 27, 28, 29, 30, 30, 31, 32, 33, 33, 34, 35, 36, 36, 37, 37, 38, 38, 39, 40, 41, 41, 42, 43, 44, 44, 45, 46, 47, 47, 48, 48, 49, 49, 50, 51, 52, 52, 53, 54

Offset: 0

Eric W. Weisstein

Note that the first equation implies a(0)=0, so there is no need to specify an initial value.
Maximal cardinality of a double-free subset of {1, 2, ..., n}, or in other words, maximal size of a subset S of {1, 2, ..., n} with the property that if x is in S then 2x is not. a(0)=0 by convention.
Least k such that a(k)=n is equal to A003159(n).
To construct the sequence: let [a, b, c, a, a, a, b, c, a, b, c, ...] be the fixed point of the morphism a -> abc, b ->a, c -> a, starting from a(1) = a, then write the indices of a, b, c, that of a being written twice; see A092606. - Philippe Deléham, Apr 13 2004
Number of integers from {1,...,n} for which the subtraction of 1 changes the parity of the number of 1's in their binary expansion. - Vladimir Shevelev, Apr 15 2010
Number of integers from {1,...,n} the factorization of which over different terms of A050376 does not contain 2. - Vladimir Shevelev, Apr 16 2010
a(n) modulo 2 is the Prouhet-Thue-Morse sequence A010060. Each number n appears A026465(n+1) times. - Philippe Deléham, Oct 19 2011
Another way of stating the last two comments from Philippe Deléham: the sequence can be obtained by replacing each term of the Thue-Morse sequence A010060 by the run number that term is in. - N. J. A. Sloane, Dec 31 2013

			Examples for n = 1 through 8: {1}, {1}, {1,3}, {1,3,4}, {1,3,4,5}, {1,3,4,5}, {1,3,4,5,7}, {1,3,4,5,7}.
Binary expansion of 5 is 101, so Sum{i>=0} b_i*(-1)^i = 2. Therefore a(5) = 10/3 + 2/3 = 4. - _Vladimir Shevelev_, Apr 15 2010

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 2.26.
Wang, E. T. H. "On Double-Free Sets of Integers." Ars Combin. 28, 97-100, 1989.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Steven R. Finch, Triple-Free Sets of Integers [From Steven Finch, Apr 20 2019]
Ralf Stephan, Some divide-and-conquer sequences ...
Ralf Stephan, Table of generating functions
Eric Weisstein's World of Mathematics, Double-Free Set.

Bisection of A123087.

Cf. A001045, A050291-A050296, A050321, A035263, A121701, A030308, A080277, A120385, A197911.

a050292 n = a050292_list !! (n-1)
a050292_list = scanl (+) 0 a035263_list
-- Reinhard Zumkeller, Jan 21 2013

A050292:=n->add((-1)^k*floor(n/2^k), k=0..n); seq(A050292(n), n=0..100); # Wesley Ivan Hurt, Feb 14 2014

a[n_] := a[n] = If[n < 2, 1, n - a[Floor[n/2]]]; Table[ a[n], {n, 1, 75}]
Join[{0},Accumulate[Nest[Flatten[#/.{0->{1,1},1->{1,0}}]&,{0},7]]] (* Harvey P. Dale, Apr 29 2018 *)

PARI
```
a(n)=if(n<2,1,n-a(floor(n/2)))
```

from sympy.ntheory import digits
def A050292(n): return ((n<<1)+sum((0,1,-1,0)[i] for i in digits(n,4)[1:]))//3 # Chai Wah Wu, Jan 30 2025

Partial sums of A035263. Close to (2/3)*n.
a(n) = A123087(2*n) = n - A123087(n). - Max Alekseyev, Mar 05 2023
From Benoit Cloitre, Nov 24 2002: (Start)
a(1)=1, a(n) = n - a(floor(n/2));
a(n) = (2/3)*n + (1/3)*A065359(n);
more generally, for m>=0, a(2^m*n) - 2^m*a(n) = A001045(m)*A065359(n) where A001045(m) = (2^m - (-1)^m)/3 is the Jacobsthal sequence;
a(A039004(n)) = (2/3)*A039004(n);
a(2*A039004(n)) = 2*a(A039004(n));
a(A003159(n)) = n;
a(A003159(n)-1) = n-1;
a(n) mod 2 = A010060(n) the Thue-Morse sequence;
a(n+1) - a(n) = A035263(n+1);
a(n+2) - a(n) = abs(A029884(n)).
(End)
G.f.: (1/(x-1)) * Sum_{i>=0} (-1)^i*x^(2^i)/(x^(2^i)-1). - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Feb 17 2003
a(n) = Sum_{k>=0} (-1)^k*floor(n/2^k). - Benoit Cloitre, Jun 03 2003
a(A091785(n)) = 2n; a(A091855(n)) = 2n-1. - Philippe Deléham, Mar 26 2004
a(2^n) = (2^(n+1) + (-1)^n)/3. - Vladimir Shevelev, Apr 15 2010
If n = Sum_{i>=0} b_i*2^i is the binary expansion of n, then a(n) = 2n/3 + (1/3)Sum_{i>=0} b_i*(-1)^i. Thus a(n) = 2n/3 + O(log(n)). - Vladimir Shevelev, Apr 15 2010
Moreover, the equation a(3m)=2m has infinitely many solutions, e.g., a(3*2^k)=2*2^k; on the other hand, a((4^k-1)/3)=(2*(4^k-1))/9+k/3, i.e., limsup |a(n)-2n/3| = infinity. - Vladimir Shevelev, Feb 23 2011
a(n) = Sum_{k>=0} A030308(n,k)*A001045(k+1). - Philippe Deléham, Oct 19 2011
From Peter Bala, Feb 02 2013: (Start)
Product_{n >= 1} (1 + x^((2^n - (-1)^n)/3 )) = (1 + x)^2(1 + x^3)(1 + x^5)(1 + x^11)(1 + x^21)... = 1 + sum {n >= 1} x^a(n) = 1 + 2x + x^2 + x^3 + 2x^4 + 2x^5 + .... Hence this sequence lists the numbers representable as a sum of distinct Jacobsthal numbers A001045 = [1, 1', 3, 5, 11, 21, ...], where we distinguish between the two occurrences of 1 by writing them as 1 and 1'. For example, 9 occurs twice in the present sequence because 9 = 5 + 3 + 1 and 9 = 5 + 3 + 1'. Cf. A197911 and A080277. See also A120385.
(End)

Extended with formula by Christian G. Bower, Sep 15 1999
Corrected and extended by Reinhard Zumkeller, Aug 16 2006
Extended with formula by Philippe Deléham, Oct 19 2011
Entry revised to give a simpler definition by N. J. A. Sloane, Jan 03 2014

cp's OEIS Frontend

A096268 Period-doubling sequence (or period-doubling word): fixed point of the morphism 0 -> 01, 1 -> 00.

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A050292 a(2n) = 2n - a(n), a(2n+1) = 2n + 1 - a(n) (for n >= 0).

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A186444 The count of numbers <= n for which 3 is an infinitary divisor.

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