A036577 -id:A036577 - 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

A007413 A squarefree (or Thue-Morse) ternary sequence: closed under 1->123, 2->13, 3->2. Start with 1.

Original entry on oeis.org

1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3

Offset: 1

N. J. A. Sloane

a(n)=2 if and only if n-1 is in A079523. - Benoit Cloitre, Mar 10 2003
Partial sums modulo 4 of the sequence 1, a(1), a(1), a(2), a(2), a(3), a(3), a(4), a(4), a(5), a(5), a(6), a(6), ... - Philippe Deléham, Mar 04 2004
To construct the sequence: start with 1 and concatenate 4 -1 = 3: 1, 3, then change the last term (2 -> 1, 3 ->2 ) gives 1, 2. Concatenate 1, 2 with 4 -1 = 3, 4 - 2 = 2: 1, 2, 3, 2 and change the last term: 1, 2, 3, 1. Concatenate 1, 2, 3, 1 with 4 - 1 = 3, 4 - 2 = 2, 4 - 3 = 1, 4 - 1 = 3: 1, 2, 3, 1, 3, 2, 1, 3 and change the last term: 1, 2, 3, 1, 3, 2, 1, 2 etc. - Philippe Deléham, Mar 04 2004
To construct the sequence: start with the Thue-Morse sequence A010060 = 0, 1, 1, 0, 1, 0, 0, 1, ... Then change 0 -> 1, 2, 3,  and 1 -> 3, 2, 1,  gives: 1, 2, 3, , 3, 2, 1, ,3, 2, 1, , 1, 2, 3, , 3, 2, 1, , ... and fill in the successive holes with the successive terms of the sequence itself. - _Philippe Deléham, Mar 04 2004
To construct the sequence: to insert the number 2 between the A003156(k)-th term and the (1 + A003156(k))-th term of the sequence 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, 1, 3, ... - Philippe Deléham, Mar 04 2004
Conjecture. The sequence is formed by the numbers of 1's between every pair of consecutive 2's in A076826. - Vladimir Shevelev, May 31 2009

			Here are the first 5 stages in the construction of this sequence, together with Mma code, taken from Keranen's article. His alphabet is a,b,c rather than 1,2,3.
productions = {"a" -> "abc ", "b" -> "ac ", "c" -> "b ", " " -> ""};
NestList[g, "a", 5] // TableForm
a
abc
abc ac b
abc ac b abc b ac
abc ac b abc b ac abc ac b ac abc b
abc ac b abc b ac abc ac b ac abc b abc ac b abc b ac abc b abc ac b ac

Michel Rigo, Formal Languages, Automata and Numeration Systems, 2 vols., Wiley, 2014. Mentions this sequence - see "List of Sequences" in Vol. 2.
J. Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 18.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. Thue, Über unendliche Zeichenreihen, Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. Christiania, No. 7 (1906), 1-22.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Roger L. Bagula, Description of sequence as successive rows of a triangle
James D. Currie, Non-repetitive words: Ages and essences, Combinatorica 16.1 (1996): 19-40. See p. 20.
James D. Currie, Palindrome positions in ternary square-free words, Theoretical Computer Science, 396 (2008) 254-257.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
V. Keranen, New Abelian Square-Free DT0L-Languages over 4 Letters, Theoretical Computer Science, Volume 410, Issues 38-40, 6 September 2009, Pages 3893-3900.
S. Kitaev and T. Mansour, Counting the occurrences of generalized patterns in words generated by a morphism, arXiv:math/0210170 [math.CO], 2002.
Andrzej Tomski and Maciej Zakarczemny, A note on Browkin's and Cao's cancellation algorithm, Technical Transections 7/2018.

Cf. A001285, A010060.
First differences of A000069.
Equals A036580(n-1) + 1.
Cf. A115384, A159481, A079523, A000120.

Nest[ Flatten[ # /. {1 -> {1, 2, 3}, 2 -> {1, 3}, 3 -> {2}}] &, {1}, 7] (* Robert G. Wilson v, May 07 2005 *)
2 - Differences[ThueMorse[Range[0, 100]]] (* Paolo Xausa, Oct 25 2024 *)

{a(n) = if( n<1 || valuation(n, 2)%2, 2, 2 + (-1)^subst( Pol(binary(n)), x,1))};

def A007413(n): return 2-(n.bit_count()&1)+((n-1).bit_count()&1) # Chai Wah Wu, Mar 03 2023

a(n) modulo 2 = A035263(n). a(A036554(n)) = 2. a(A003159(n)) = 1 if n odd. a(A003159(n)) = 3 if n even. a(n) = A033485(n) mod 4. a(n) = 4 - A036585(n-1). - Philippe Deléham, Mar 04 2004
a(n) = 2 - A029883(n) = 3 - A036577(n). - Philippe Deléham, Mar 20 2004
For n>=1, we have: 1) a(A108269(n))=A010684(n-1); 2) a(A079523(n))=A010684(n-1); 3) a(A081706(2n))=A010684(n). - Vladimir Shevelev, Jun 22 2009

A316826 Image of 3 under repeated application of the morphism 3 -> 3,2, 2 -> 1,0,2,0,1,2, 1 -> 1,0,1,2, 0 -> 0,2.

Original entry on oeis.org

3, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0

Offset: 0

N. J. A. Sloane, Jul 14 2018

nonn

A word that is pure morphic and uniform morphic, but neither pure uniform morphic, nor primitive morphic, nor recurrent.

Robert Israel, Table of n, a(n) for n = 0..10000
Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017.

Sequences mentioned in the Allouche et al. "Taxonomy" paper, listed by example number: 1: A003849, 2: A010060, 3: A010056, 4: A020985 and A020987, 5: A191818, 6: A316340 and A273129, 18: A316341, 19: A030302, 20: A063438, 21: A316342, 22: A316343, 23: A003849 minus its first term, 24: A316344, 25: A316345 and A316824, 26: A020985 and A020987, 27: A316825, 28: A159689, 29: A049320, 30: A003849, 31: A316826, 32: A316827, 33: A316828, 34: A316344, 35: A043529, 36: A316829, 37: A010060.

Cf. A036577.

S:= [3=(3,2), 2 = (1,0,2,0,1,2), 1 = (1,0,1,2), 0 = (0,2)]:
A:= [3]:
for iter from 1 do
  Ap:= subs(S,A);
  if nops(Ap) > 100 then Ap:= Ap[1..100] fi;
  if Ap = A then break fi;
  A:= Ap
od:
A; # Robert Israel, Jul 30 2020

SubstitutionSystem[{3 -> {3, 2}, 2 -> {1, 0, 2, 0, 1, 2}, 1 -> {1, 0, 1, 2}, 0 -> {0, 2}}, {3}, 4] // Last (* Jean-François Alcover, Nov 11 2018 *)

A316344 An example of a word that is uniform morphic, but neither pure morphic, primitive morphic, nor recurrent.

Original entry on oeis.org

2, 2, 0, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2

Offset: 0

N. J. A. Sloane, Jul 14 2018

Zhuorui He, Table of n, a(n) for n = 0..10000 (first 1000 terms from Jack W Grahl)
Jean-Paul Allouche, Julien Cassaigne, Jeffrey Shallit, and Luca Q. Zamboni, A Taxonomy of Morphic Sequences, arXiv preprint arXiv:1711.10807 [cs.FL], Nov 29 2017. See Example 24.
Jack W Grahl, Haskell code to generate this sequence

Cf. A036577.

Join[{2, 2}, Differences[ThueMorse[Range[2, 100]]] + 1] (* Paolo Xausa, Jul 17 2025 *)

From Zhuorui He, Jul 11 2025: (Start)
a(n) = A010060(2*n+2) + A010060(max(2*n+1,4)).
a(n) = A036577(n+1) except a(1) = 2. (End)

More terms from Jack W Grahl, Jul 23 2018

A091855 Odious numbers (see A000069) in A003159.

Original entry on oeis.org

1, 4, 7, 11, 13, 16, 19, 21, 25, 28, 31, 35, 37, 41, 44, 47, 49, 52, 55, 59, 61, 64, 67, 69, 73, 76, 79, 81, 84, 87, 91, 93, 97, 100, 103, 107, 109, 112, 115, 117, 121, 124, 127, 131, 133, 137, 140, 143, 145, 148, 151, 155, 157, 161, 164, 167, 171, 173, 176, 179, 181

Offset: 1

Philippe Deléham, Mar 16 2004

Also n such that A033485(n) == 1 (mod 4); see A007413.
Also n such that A029883(n-1) = 1, A036577(n-1) = 2, A036585(n-1) = 3. - Philippe Deléham, Mar 25 2004
The number of odd numbers before the n-th even number in this sequence is a(n). - Philippe Deléham, Mar 27 2004
Numbers n such that {A010060(n-1), A010060(n)}={0,1} where A010060 is the Thue-Morse sequence. - Benoit Cloitre, Jun 16 2006
Positive integers not of the form n+A010060(n). - Jeffrey Shallit, Feb 13 2024

G. C. Greubel, Table of n, a(n) for n = 1..10000
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), no. 1, 42-46.
Index entries for 2-automatic sequences.

A036554 := Select[Range[70500], OddQ[IntegerExponent[#, 2]] &]; Table[ A036554[[2*n - 1]]/2, {n, 1, 100}] (* G. C. Greubel, Jan 14 2018 *)

is(n)=hammingweight(n)%2 && valuation(n,2)%2==0 \\ Charles R Greathouse IV, May 09 2016

a(n) = A003159(2n-1) = A036554(2n-1)/2.
a(n) is asymptotic to 3*n - Benoit Cloitre, Mar 22 2004
A050292(a(n)) = 2n - 1. - Philippe Deléham, Mar 26 2004

More terms from Benoit Cloitre, Mar 22 2004

A029883 First differences of Thue-Morse sequence A001285.

Original entry on oeis.org

1, 0, -1, 1, -1, 0, 1, 0, -1, 0, 1, -1, 1, 0, -1, 1, -1, 0, 1, -1, 1, 0, -1, 0, 1, 0, -1, 1, -1, 0, 1, 0, -1, 0, 1, -1, 1, 0, -1, 0, 1, 0, -1, 1, -1, 0, 1, -1, 1, 0, -1, 1, -1, 0, 1, 0, -1, 0, 1, -1, 1, 0, -1, 1, -1, 0, 1, -1, 1, 0, -1, 0, 1, 0, -1, 1, -1, 0, 1, -1, 1, 0, -1, 1, -1, 0, 1, 0, -1, 0, 1, -1, 1, 0, -1, 0, 1, 0, -1, 1, -1, 0, 1, 0, -1

Offset: 1

N. J. A. Sloane, Dec 11 1999

Also first differences of {0,1} Thue-Morse sequence A010060.- N. J. A. Sloane, Jan 05 2021
Fixed point of the morphism a->abc, b->ac, c->b, with a = 1, b = 0, c = -1, starting with a(1) = 1. - Philippe Deléham
From Thomas Anton, Sep 22 2020: (Start)
This sequence, interpreted as an infinite word, is squarefree.
Let & represent concatenation. For a word w of integers, let -w be the same word with each symbol negated. Then, starting with the empty word, this sequence can be obtained by iteratively applying the transformation T(w) = w & 1 & -w & 0 & -w & -1 & w. (End)

G. C. Greubel, Table of n, a(n) for n = 1..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.
G. N. Arzhantseva, C. H. Cashen, D. Gruber, and D. Hume, Contracting geodesics in infinitely presented graphical small cancellation groups, arXiv preprint arXiv:1602.03767 [math.GR], 2016-2017.
T. W. Cusick, H. Fredricksen and P. Stănică, On the delta sequence of the Thue-Morse sequence, Australas. J. Combin. 39 (2007), 293--300. [From _N. J. A. Sloane_, Dec 11 2009]
Florian Frohn and Jürgen Giesl, Proving Non-Termination by Acceleration Driven Clause Learning with LoAT, arXiv:2304.10166 [cs.LO], 2023.
Index entries for sequences that are fixed points of mappings

Apart from signs, same as A035263. Cf. A001285, A010060, A036554, A091785, A091855.

a(n+1) = A036577(n) - 1 = A036585(n) - 2.

Nest[ Function[ l, {Flatten[(l /. {0 -> {1, -1}, 1 -> {1, 0, -1}, -1 -> {0}})]}], {1}, 7] (* Robert G. Wilson v, Feb 26 2005 *)
ThueMorse /@ Range[0, 105] // Differences (* Jean-François Alcover, Oct 15 2019 *)

a(n)=if(n<1||valuation(n,2)%2,0,-(-1)^subst(Pol(binary(n)),x,1)) /* Michael Somos, Jul 08 2004 */

a(n)=hammingweight(n)%2-hammingweight(n-1)%2 \\ Charles R Greathouse IV, Mar 26 2013

def A029883(n): return (bin(n).count('1')&1)-(bin(n-1).count('1')&1) # Chai Wah Wu, Mar 03 2023

Recurrence: a(4*n) = a(n), a(4*n+1) = a(2*n+1), a(4*n+2) = 0, a(4*n+3) = -a(2*n+1), starting a(1) = 1.
a(n) = 2 - A007413(n). a(A036554(n)) = 0; a(A091785(n)) = -1; a(A091855(n)) = 1. - Philippe Deléham, Mar 20 2004
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = -v+w+u^2-v^2+2*w^2-2*u*w. - Michael Somos, Jul 08 2004

Edited by Ralf Stephan, Dec 09 2004

A036585 Ternary Thue-Morse sequence: closed under a->abc, b->ac, c->b.

Original entry on oeis.org

3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 1, 3, 2, 1, 3, 1, 2, 3, 2, 1, 2, 3, 1, 3, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1

Offset: 1

N. J. A. Sloane

nonn

First differences of A001969. Observed by Franklin T. Adams-Watters, proved by Max Alekseyev, Aug 30 2006

M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 26.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A001969, A007413, A005679.

a036585 n = a036585_list !! (n-1)
a036585_list = 3 : concat (map f a036585_list)
where f 1 = [1,2,3]; f 2 = [1,3]; f 3 = [2]
-- Reinhard Zumkeller, Oct 31 2012

Differences[ThueMorse[Range[0, 100]]] + 2 (* Paolo Xausa, Oct 25 2024 *)

a(n)=if(n<1 || valuation(n,2)%2,2,2-(-1)^subst(Pol(binary(n)),x,1))

def A036585(n): return 2+(n.bit_count()&1)-((n-1).bit_count()&1) # Chai Wah Wu, Mar 03 2023

a(n) = A001969(n+1) - A001969(n). - Franklin T. Adams-Watters, Aug 28 2006

a(n) = A029883(n) + 2 = A036577(n) + 1.

A036580 Ternary Thue-Morse sequence: closed under a->abc, b->ac, c->b.

Original entry on oeis.org

0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 2, 0, 2, 1, 0, 2, 0, 1, 2, 0, 2, 1, 0, 1, 2, 1, 0, 2, 0, 1, 2, 1, 0, 1, 2

Offset: 0

N. J. A. Sloane

nonn

0 is a, 1 is b and 2 is c. - Robert G. Wilson v, Jul 30 2018

M. Lothaire, Combinatorics on Words. Addison-Wesley, Reading, MA, 1983, p. 26.

Paolo Xausa, Table of n, a(n) for n = 0..10000
Marko Milosevic and Narad Rampersad, Squarefree words with interior disposable factors, arXiv:2007.03557 [math.CO], 2020.
Michaël Rao, Michel Rigo, and Pavel Salimov, Avoiding 2-binomial squares and cubes, arXiv:1310.4743 [cs.FL], 2013.
Michaël Rao, Michel Rigo, and Pavel Salimov, Avoiding 2-binomial squares and cubes, Theoretical Computer Science, Volume 572, 23 March 2015, Pages 83-91.

A007413(n+1) - 1.

See A036577 for another version.

Nest[Flatten[# /. {0 -> {0, 1, 2}, 1 -> {0, 2}, 2 -> {1}}] &, {1}, 7] // Most (* Robert G. Wilson v, Jul 30 2018, corrected by Paolo Xausa, Jul 21 2025 *)
1 - Differences[ThueMorse[Range[0,100]]] (* Paolo Xausa, Jul 21 2025 *)

A091785 Evil numbers (see A001969) in A003159.

Original entry on oeis.org

3, 5, 9, 12, 15, 17, 20, 23, 27, 29, 33, 36, 39, 43, 45, 48, 51, 53, 57, 60, 63, 65, 68, 71, 75, 77, 80, 83, 85, 89, 92, 95, 99, 101, 105, 108, 111, 113, 116, 119, 123, 125, 129, 132, 135, 139, 141, 144, 147, 149, 153, 156, 159, 163, 165, 169, 172, 175, 177, 180, 183

Offset: 1

Philippe Deléham, Mar 16 2004

Also n such that A033485(n) == 3 (mod 4); see A007413.
Also n such that A029883(n-1) = -1, A036577(n-1) = 0, A036585(n-1) = 1. - Philippe Deléham, Mar 25 2004
The number of odd numbers before the n-th even number in this sequence is a(n). - Philippe Deléham, Mar 27 2004
Numbers n such that {A010060(n-1), A010060(n)}={1,0} where A010060 is the Thue-Morse sequence. - Benoit Cloitre, Jun 16 2006

G. C. Greubel, Table of n, a(n) for n = 1..10000
Aviezri S. Fraenkel, The vile, dopey, evil and odious game players, Discrete Math. 312 (2012), pp. 42-46.
Index entries for 2-automatic sequences.

A036554 := Select[Range[750], OddQ[IntegerExponent[#, 2]] &]; Table[ A036554[[2*n]]/2, {n, 1, 50}] (* G. C. Greubel, Jan 14 2018 *)

is(n)=hammingweight(n-1)%2 && hammingweight(n)%2==0 \\ Charles R Greathouse IV, Mar 26 2013

a(n) = A003159(2*n) = A036554(2*n)/2.
a(n) is asymptotic to 3*n. - Benoit Cloitre, Mar 22 2004
A050292(a(n)) = 2n. - Philippe Deléham, Mar 26 2004

More terms from Benoit Cloitre, Mar 22 2004

cp's OEIS Frontend

A216199 Abelian complexity function of the ternary Thue-Morse word (A036577).

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

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A007413 A squarefree (or Thue-Morse) ternary sequence: closed under 1->123, 2->13, 3->2. Start with 1.

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A316826 Image of 3 under repeated application of the morphism 3 -> 3,2, 2 -> 1,0,2,0,1,2, 1 -> 1,0,1,2, 0 -> 0,2.

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A316344 An example of a word that is uniform morphic, but neither pure morphic, primitive morphic, nor recurrent.

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A091855 Odious numbers (see A000069) in A003159.

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A029883 First differences of Thue-Morse sequence A001285.

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