A001285 -id:A001285 - cp's OEIS frontend

A003324 A nonrepetitive sequence.

1, 2, 3, 4, 1, 4, 3, 2, 1, 2, 3, 2, 1, 4, 3, 4, 1, 2, 3, 4, 1, 4, 3, 4, 1, 2, 3, 2, 1, 4, 3, 2, 1, 2, 3, 4, 1, 4, 3, 2, 1, 2, 3, 2, 1, 4, 3, 2, 1, 2, 3, 4, 1, 4, 3, 4, 1, 2, 3, 2, 1, 4, 3, 4, 1, 2, 3, 4, 1

Offset: 1

N. J. A. Sloane

Let b(0) be the sequence 1,2,3,4. Proceeding by induction, let b(n) be a sequence of length 2^(n+2). Quarter b(n) into four blocks, A,B,C,D each of length 2^n, so that b(n) = ABCD. Then b(n+1) = ABCDADCB. [After Dean paper.] - Sean A. Irvine, Apr 20 2015

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Sean A. Irvine, Table of n, a(n) for n = 1..10000
Richard A. Dean, A sequence without repeats on x, x^{-1}, y, y^{-1}, Amer. Math. Monthly 72, 1965. pp. 383-385. MR 31 #350.
Françoise Dejean, Sur un Théorème de Thue, J. Combinatorial Theory, vol. 13 A, iss. 1 (1972) 90-99.
N. J. A. Sloane, P. Flor, L. F. Meyers, G. A. Hedlund. M. Gardner, Collection of documents and notes related to A1285, A3270, A3324
Jianing Song, Proof for my formula for A003324

Positions of 1's, 2's, 3's and 4's: A016813, A343500, A004767, A343501.

Cf. A292077.

b[0] = Range[4];
b[n_] := b[n] = Module[{aa, bb, cc, dd}, {aa, bb, cc, dd} = Partition[b[n - 1], 2^(n-1)]; Join[aa, bb, cc, dd, aa, dd, cc, bb] // Flatten];
b[5] (* Jean-François Alcover, Sep 27 2017 *)
a[n_] := If[OddQ[n], Mod[n, 4], Module[{e = IntegerExponent[n, 2], k}, k = (n/2^e - 1)/2; If[OddQ[k + e], 2, 4]]];
Array[a, 100] (* Jean-François Alcover, Apr 19 2021, after Jianing Song *)

a(n) = if(n%2, n%4, my(e=valuation(n,2), k=bittest(n, e+1)); if((k+e)%2, 2, 4)) \\ Jianing Song, Apr 15 2021

def A003324(n): return n&3 if n&1 else 2<<(((n>>(m:=(~n&n-1).bit_length()))+1>>1)+m&1) # Chai Wah Wu, Feb 26 2025

a(n) = n mod 4 for odd n; for even n, write n = (2*k+1) * 2^e, then a(n) = 2 if k+e is odd, 4 if k+e is even. - Jianing Song, Apr 15 2021

Conjecture: a(2*n) = (A292077(n)+1)*2. Confirmed for first 1000 terms. - John Keith, Apr 18 2021 [This conjecture is correct. Write n = (2*k+1) * 2^e. If k+e is even, then we have A292077(n) = 0 and a(2n) = 2; if k+e is odd, then we have A292077(n) = 1 and a(2n) = 4. - Jianing Song, Nov 27 2021]

A006996 a(n) = C(2n,n) mod 3.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, James Propp

Removing 0's from the sequence gives Thue-Morse sequence A001285 : 1,2,0,2,1,0,0,0,0,2,1,0,1,2,..->1,2,2,1,2,1,1,2,... - Benoit Cloitre, Jan 04 2004
a(n) = 0 if n in A074940, a(n) = 1 if n in A074939, a(n) = 2 if n in A074938.
Central terms of the triangle in A083093. - Reinhard Zumkeller, Jul 11 2013

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Reinhard Zumkeller, Table of n, a(n) for n = 0..2187=3^7
Michael Gilleland, Some Self-Similar Integer Sequences
Index entries for sequences that are fixed points of mappings

Cf. A001285.
Cf. A074938, A074939, A074940.
Cf. A083093.
Cf. A000984, A005704.

a006996 n = a083093 (2 * n) n  -- Reinhard Zumkeller, Jul 11 2013

Table[ Mod[ Binomial[2n, n], 3], {n, 0, 104}] (* Or *)
Nest[ Function[ l, {Flatten[(l /. {0 -> {0, 0, 0}, 1 -> {1, 2, 0}, 2 -> {2, 1, 0}})]}], {1}, 7] (* Robert G. Wilson v, Mar 28 2005 *)

a(n)=if(n==0, return(1)); if(vecmax(Set(digits(n,3)))>1, 0, 1 + n%2) \\ Charles R Greathouse IV, May 09 2016

from gmpy2 import digits
def A006996(n): return 0 if '2' in digits(n,3) else 1+(n&1) # Chai Wah Wu, Jun 26 2025

a(n) = A000984(n) mod 3.
a(n) = A005704(n) mod 3. - Benoit Cloitre, Jan 04 2004
A fixed point of the morphism : 1 -> 120, 2 -> 210, 0 -> 000. - Philippe Deléham, Jan 08 2004

A064990 If A_k denotes the first 3^k terms, then A_0 = 0, A_{k+1} = A_k A_k B_k, where B_k is obtained from A_k by interchanging 0's and 1's.

Original entry on oeis.org

0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0

Offset: 0

Michael Gilleland (megilleland(AT)yahoo.com), Oct 31 2001

Called the Mephisto Waltz sequence (or the Mephisto Waltz infinite word).
May also be obtained by starting with 0 and iterating the morphism 0 -> 001, 1 -> 110.
The sequence is fourth-power free.
The sequence gives A_oo. For the concatenation A_0, A_1, A_2, ... see A134391.

			Here are A_0 through A_4:
  0
  001
  001001110
  001001110001001110110110001
  001001110001001110110110001001001110001001110110110001110110001110110001001001110

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 25.
Konrad Jacobs, Invitation to Mathematics, Princeton, 1992; pp. 105-106 and 215.

Joerg Arndt, Matters Computational (The Fxtbook), section 38.1.2, pp. 729-730
J. Endrullis, D. Hendriks and J. W. Klop, Degrees of streams.
Joseph Meleshko, Pascal Ochem, Jeffrey Shallit, and Sonja Linghui Shan, Pseudoperiodic Words and a Question of Shevelev, arXiv:2207.10171 [math.CO], 2022.
Eric Weisstein's World of Mathematics, Mephisto Waltz Sequence

Cf. Thue-Morse sequence A010060, A001285. Number of 0's in A_k gives A007051, number of 1's is A003462. See also A064991.
Cf. A134391, A189628.
A285196 is a similar sequence.

with(ListTools);
f2:=proc(S) map(x->x+1 mod 2, S); end;
f:=proc(S) global f2;
[op(S), op(S), op(f2(S))]; end;
S:=[0];
for n from 1 to 6 do S:=f(S): od:
S; # N. J. A. Sloane, Apr 30 2017

t = Nest[Flatten[# /. {0->{0,0,1}, 1->{1,1,0}}] &, {0}, 5] (*A064990*)
f[n_] := t[[n]]
Flatten[Position[t, 0]] (* A189658 *)
Flatten[Position[t, 1]] (* A189659 *)
s[n_] := Sum[f[i], {i, 1, n}]; s[0] = 0;
Table[s[n], {n, 1, 120}] (* A189660 *)
(* Clark Kimberling, Apr 25 2011 *)
Nest[ Flatten[# /. # -> {#, #, Abs[# - 1]}] &, {0}, 5]  (* Robert G. Wilson v, Sep 27 2011 *)
SubstitutionSystem[{0->{0,0,1},1->{1,1,0}},{0},{5}][[1]] (* Harvey P. Dale, Jan 25 2022 *)

a(n) = vecsum(digits(n,3)>>1)%2; \\ Kevin Ryde, Jun 02 2020

a(3k-2)=a(k), a(3k-1)=a(k), a(3k)=1-a(k) for k>=1, a(0)=0.

More terms from Naohiro Nomoto, Nov 29 2001

Corrected by N. J. A. Sloane, Jun 14 2010, at the suggestion of Chris Erickson

A181155 Odious numbers (A000069) plus one; complement of A026147.

Original entry on oeis.org

2, 3, 5, 8, 9, 12, 14, 15, 17, 20, 22, 23, 26, 27, 29, 32, 33, 36, 38, 39, 42, 43, 45, 48, 50, 51, 53, 56, 57, 60, 62, 63, 65, 68, 70, 71, 74, 75, 77, 80, 82, 83, 85, 88, 89, 92, 94, 95, 98, 99, 101, 104, 105, 108, 110, 111, 113, 116, 118, 119, 122, 123, 125, 128, 129, 132

Offset: 1

Matthew Vandermast, Oct 06 2010

a(n) = position of n-th 2 in A001285 if offset for A001285 is given as 1.
It appears that this sequence and A026147 index each other's even terms (i.e., a(n) = position of n-th even term in A026147, and A026147(n) = position of n-th even term in this sequence). It also appears that each of the two sequences indexes its own odd terms (cf. A079000).
Barbeau notes that if let A = the first 2^k terms of A026147 and B = the first 2^k terms of this sequence, then the two sets have the same sum of powers for first up to the k-th power. I note it holds for 0th power also. - Michael Somos, Jun 09 2013

			Let k=2. Then A = {1,4,6,7} and B = {2,3,5,8} have the property that 1^0+4^0+6^0+7^0 = 2^0+3^0+5^0+8^0 = 4, 1^1+4^1+6^1+7^1 = 2^1+3^1+5^1+8^1 = 18, and 1^2+4^2+6^2+7^2 = 2^2+3^2+5^2+8^2 = 102. - _Michael Somos_, Jun 09 2013

G. C. Greubel, Table of n, a(n) for n = 1..1000
Edward J. Barbeau, Power Play, MAA, 1997. See p. 104.

Cf. A026147.

a[ n_] := If[ n < 1, 0, 2 n - Mod[ Total[ IntegerDigits[ n - 1, 2]], 2]] (* Michael Somos, Jun 09 2013 *)

a(n)=2*n-hammingweight(n-1)%2 \\ Charles R Greathouse IV, Mar 22 2013

{a(n) = if( n<1, 0, 2*n - subst( Pol( binary( n-1)), x, 1)%2)} /* Michael Somos, Jun 09 2013 */

def A181155(n): return 1+((m:=n-1)<<1)+(m.bit_count()&1^1) # Chai Wah Wu, Mar 03 2023

a(n) = A000069(n) + 1.
a(a(n)-1) = 2*a(n)-1. - Benoit Cloitre, Oct 07 2010
a(n) + A010060(n+1) = 2n + 2  for n >= 0. - Clark Kimberling, Oct 06 2014

A071858 (Number of 1's in binary expansion of n) mod 3.

Original entry on oeis.org

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

Offset: 0

Benoit Cloitre, Jun 09 2002

This is the generalized Thue-Morse sequence t_3 (Allouche and Shallit, p. 335).
Ternary sequence which is a fixed point of the morphism 0 -> 01, 1 -> 12, 2 -> 20.
Sequence is T^(oo)(0) where T is the operator acting on any word on alphabet {0,1,2} by inserting 1 after 0, 2 after 1 and 0 after 2. For instance T(001)=010112, T(120)=122001. - Benoit Cloitre, Mar 02 2009

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Jin Chen, Zhixiong Wen, Wen Wu, On the additive complexity of a Thue-Morse like sequence, arXiv:1802.03610 [math.CO], 2018.
Index entries for sequences that are fixed points of mappings

Cf. A000120, A010872.
Cf. A010060, A001285, A010059, A048707, A096271, A100619, A179868.
See A245555 for another version.

f[n_] := Mod[ Count[ IntegerDigits[n, 2], 1], 3]; Table[ f[n], {n, 0, 104}] (* Or *)
Nest[ Flatten[ # /. {0 -> {0, 1}, 1 -> {1, 2}, 2 -> {2, 0}}] &, {0}, 7] (* Robert G. Wilson v Mar 03 2005, modified May 17 2014 *)
Table[Mod[DigitCount[n,2,1],3],{n,0,110}] (* Harvey P. Dale, Jul 01 2015 *)

for(n=1,200,print1(sum(i=1,length(binary(n)), component(binary(n),i))%3,","))

map(d)=if(d==2,[2,0],if(d==1,[1,2],[0,1]))
{m=53;v=[];w=[0];while(v!=w,v=w;w=[];for(n=1,min(m,length(v)),w=concat(w,map(v[n]))));for(n=1,2*m,print1(v[n],","))} \\ Klaus Brockhaus, Jun 23 2004

a(n) = A010872(A000120(n)).
Recurrence: a(2*n) = a(n), a(2*n+1) = (a(n)+1) mod 3.
a(n) = A000695(n) mod 3. - John M. Campbell, Jul 16 2016

Edited by Ralf Stephan, Dec 11 2004

A005679 A squarefree (or Thue-Morse) ternary sequence: closed under a->abc, b->ac, c->b.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Fixed point of the morphism 1 -> 23, 2 -> 213 & 3 -> 1. - Robert G. Wilson v, Apr 06 2008

Replacing all 4's in A005681 with 1's yields this sequence. - Sean A. Irvine, Aug 04 2016

A. Salomaa, Jewels of Formal Language Theory. Computer Science Press, Rockville, MD, 1981, p. 10.
G. Siebert, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Marston Morse and Gustav A. Hedlund, Unending chess, symbolic dynamics and a problem in semigroups, Duke Math. J., Volume 11, Number 1 (1944), 1-7.
G. Siebert, Letter to N. J. A. Sloane, Sept. 1977.
Index entries for sequences that are fixed points of mappings

Cf. A005681.

Nest[ # /. {1 -> {2, 3}, 2 -> {2, 1, 3}, 3 -> 1} &, {2}, 7] // Flatten (* Robert G. Wilson v, Apr 06 2008 *)
SubstitutionSystem[{1->{2,3},2->{2,1,3},3->{1}},{2},{7}][[1]] (* Harvey P. Dale, Jul 14 2022 *)

More terms from Robert G. Wilson v, Apr 06 2008

A104248 Lengths of successive runs of 1's in the Thue-Morse sequence A010060.

Original entry on oeis.org

2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 1, 2, 2

Offset: 1

N. J. A. Sloane, Aug 05 2008

Also lengths of successive runs of 0's in the Thue-Morse sequence A010059.
Also lengths of successive runs of 2's in the Thue-Morse sequence A001285.
A variant of A036577, suggested by p. 4421 of Grytczuk.

			A010060 begins 011010011001011010010110011010011... so the runs of 1's have lengths 2 1 2 1 2 1 1 2 2 1 2 1 2 2 1 1 2 1 ...

Ray Chandler, Table of n, a(n) for n=1..10922
Jaroslaw Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429.

Cf. A010060, A036577, A143331.

Cf. A001285, A010059, A026465.

Map[Length,Most[Split[ThueMorse[Range[500]]]][[;;;;2]]] (* Paolo Xausa, Dec 19 2023 *)
Length/@DeleteCases[Split[ThueMorse[Range[450]]],?(#[[1]]==0&)] (* _Harvey P. Dale, Nov 09 2024 *)

def A104248(n):
    def iterfun(f,n=0):
        m, k = n, f(n)
        while m != k: m, k = k, f(k)
        return m
    def f(x):
        c, s = x, bin(x)[2:]
        l = len(s)
        for i in range(l&1^1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    return iterfun(lambda x:f(x)+(n<<1),n<<1)-iterfun(lambda x:f(x)+(n<<1)-1,(n<<1)-1) # Chai Wah Wu, Jan 30 2025

a(n) = A026465(2n).

Edited and extended by Ray Chandler, Aug 08 2008

A005681 A squarefree quaternary sequence.

Original entry on oeis.org

2, 4, 3, 2, 3, 1, 2, 4, 3, 1, 2, 3, 2, 4, 3, 2, 3, 1, 2, 3, 2, 4, 3, 1, 2, 4, 3, 2, 3, 1, 2, 4, 3, 1, 2, 3, 2, 4, 3, 1, 2, 4, 3, 2, 3, 1, 2, 3, 2, 4, 3, 2, 3, 1, 2, 4, 3, 1, 2, 3, 2, 4, 3, 2, 3, 1, 2, 3, 2, 4, 3, 1, 2, 4, 3, 2, 3, 1, 2, 3, 2, 4, 3, 2, 3, 1, 2, 4, 3, 1, 2, 3, 2, 4, 3, 1, 2, 4, 3, 2, 3, 1, 2, 4, 3, 1, 2, 3, 2, 4, 3, 2, 3, 1, 2

Offset: 1

N. J. A. Sloane

nonn

This is an automatic sequence, fixed point starting with 2 of the morphism mu: 1->23, 2->24, 3->31, 4->32. The morphism mu is the 2-block morphism of the Thue-Morse morphism. An instance of this sequence on the alphabet {0,1,2,3} is equal to A245188. - Michel Dekking, Feb 18 2021

A. Salomaa, Jewels of Formal Language Theory. Computer Science Press, Rockville, MD, 1981, p. 10.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Sean A. Irvine, Table of n, a(n) for n = 1..10000
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
G. Siebert, Letter to N. J. A. Sloane, Sept. 1977

Cf. A001285, A005679, A245188.

a(n) = s(A001285(n), A001285(n+1)) where s(1,1)=1, s(1,2)=2, s(2,1)=3, s(2,2)=4. - Sean A. Irvine, Aug 04 2016

A367515 The number of unitary divisors of n that are exponentially odious numbers (A270428).

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 1, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, 4, 4, 2, 2, 2, 4, 1, 4, 2, 8, 2, 1, 4, 4, 4, 4, 2, 4, 4, 2, 2, 8, 2, 4, 4, 4, 2, 4, 2, 4, 4, 4, 2, 2, 4, 2, 4, 4, 2, 8, 2, 4, 4, 1, 4, 8, 2, 4, 4, 8, 2, 2, 2, 4, 4, 4, 4, 8, 2, 4, 2, 4, 2, 8, 4, 4, 4

Offset: 1

Amiram Eldar, Nov 21 2023

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001285, A262675, A270428, A293439.

Similar sequences: A034444, A055076, A056624, A366901, A366902, A367516.

f[p_, e_] := If[OddQ[DigitCount[e, 2, 1]], 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> hammingweight(x)%2+1, factor(n)[, 2]));

from sympy import factorint
def A367515(n): return 1<Chai Wah Wu, Nov 23 2023

Multiplicative with a(p^e) = A001285(e).
a(n) = A034444(n)/A367516(n).
a(n) = 2^A293439(n).
a(n) >= 1, with equality if and only if n is an exponentially evil number (A262675).
a(n) <= A034444(n), with equality if and only if n is an exponentially odious number (A270428).

A373308 Expansion of Product_{n>=0} (1 - x^(2^n))^3.

Original entry on oeis.org

1, -3, 0, 8, -9, 3, 8, -24, 15, 19, -24, 0, 17, -27, 0, 64, -57, -21, 64, -72, 33, 53, -72, 24, 17, -51, 24, 64, -81, 27, 64, -192, 135, 107, -192, 120, 1, -171, 120, 152, -183, -27, 152, -192, 87, 163, -192, 0, 89, -75, 0, 136, -129, -21, 136, -216, 111, 179, -216, 0, 145, -219, 0, 512

Offset: 0

Paul D. Hanna, Jun 20 2024

Equals the self-convolution cube of the signed Thue-Morse sequence A106400: [1, -1, -1, 1, -1, 1, 1, -1, ...].
Conjecture: a(3*n) == A001285(n) (mod 3) for n >= 0, where A001285 is the Thue-Morse sequence: [1, 2, 2, 1, 2, 1, 1, 2, ...].
Conjecture: a(3*n+1) and a(3*n+2) are divisible by 3 for n >= 0.
Conjecture: a(n) = 0 iff n = 3*A000695(k) - 1 for k >= 1, where A000695 lists sums of distinct powers of 4.

			G.f.: A(x) = 1 - 3*x + 8*x^3 - 9*x^4 + 3*x^5 + 8*x^6 - 24*x^7 + 15*x^8 + 19*x^9 - 24*x^10 + 17*x^12 - 27*x^13 + 64*x^15 - 57*x^16 - 21*x^17 + 64*x^18 + ...
where A(x) = (1-x)^3 * (1-x^2)^3 * (1-x^4)^3 * (1-x^8)^3 * (1-x^16)^3 * ...
Notice that a(n) = 0 at n = [2, 11, 14, 47, 50, 59, 62, 191, 194, 203, 206, 239, 242, 251, 254, 767, ...] which appears to equal 3*A000695(k) - 1 for k >= 1.

Paul D. Hanna, Table of n, a(n) for n = 0..16400

Cf. A373309, A106400, A106407, A001285, A000695.

{a(n) = my(A = prod(k=0,#binary(n), (1 - x^(2^k))^3 +x*O(x^n))); polcoeff(A, n)}
for(n=0,60, print1(a(n),", "))

G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A(x) = Product_{n>=0} (1 - x^(2^n))^3.
(2) A(x) = (1-x)^3 * A(x^2).
(3) a(n) = Sum_{k=0..n} A106400(k) * A106407(n-k+1) for n >= 0.

cp's OEIS Frontend

A003324 A nonrepetitive sequence.

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A006996 a(n) = C(2n,n) mod 3.

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Haskell

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A064990 If A_k denotes the first 3^k terms, then A_0 = 0, A_{k+1} = A_k A_k B_k, where B_k is obtained from A_k by interchanging 0's and 1's.

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Maple

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A181155 Odious numbers (A000069) plus one; complement of A026147.

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A071858 (Number of 1's in binary expansion of n) mod 3.

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A005679 A squarefree (or Thue-Morse) ternary sequence: closed under a->abc, b->ac, c->b.

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