A001285 -id:A001285 - cp's OEIS frontend

A026430 a(n) is the sum of first n terms of A001285 (Thue-Morse sequence).

0, 1, 3, 5, 6, 8, 9, 10, 12, 14, 15, 16, 18, 19, 21, 23, 24, 26, 27, 28, 30, 31, 33, 35, 36, 37, 39, 41, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 57, 59, 60, 61, 63, 65, 66, 68, 69, 70, 72, 73, 75, 77, 78, 80, 81, 82, 84, 86, 87, 88, 90, 91, 93

Offset: 0

Clark Kimberling

Winston de Greef, Table of n, a(n) for n = 0..10000 (first 1001 terms from T. D. Noe)
Nicholas John Bizzell-Browning, LIE scales: Composing with scales of linear intervallic expansion, Ph. D. Thesis, Brunel Univ. (UK, 2024). See p. 143.

Cf. A001285, A356133 (complement).

Cf. A115384.

a026430 n = a026430_list !! n
a026430_list = scanl (+) 0 a001285_list -- Reinhard Zumkeller, Jun 28 2013

A001285 = Table[ Mod[ Sum[ Mod[ Binomial[n, k], 2], {k, 0, n}], 3], {n, 0, 61}]; Accumulate[A001285] (* Jean-François Alcover, Sep 25 2012 *)
Join[{0}, Accumulate[1 + ThueMorse /@ Range[0, 100]]] (* Jean-François Alcover, Sep 18 2019, from version 10.2 *)

first(n)=my(v=vector(n)); v[1]=1; for(k=2,n,v[k]=if(k%2,v[k\2+1]-v[k\2])+k\2*3); concat(0,v) \\ Charles R Greathouse IV, May 09 2016

from itertools import accumulate, islice
def A026430_gen(): # generator of terms
    yield from (0,1)
    blist, s = [1], 1
    while True:
        c = [3-d for d in blist]
        blist += c
        yield from (s+d for d in accumulate(c))
        s += sum(c)
A026430_list = list(islice(A026430_gen(),30)) # Chai Wah Wu, Feb 22 2023

def A026430(n): return n+(n-1>>1)+(n-1&1|(n.bit_count()&1^1)) # Chai Wah Wu, Mar 01 2023

a(0)=0, a(1)=1, a(2n) = 3n, a(2n+1) = -a(n) + a(n+1) + 3n. - Ralf Stephan, Oct 08 2003

G.f.: x*(3/(1 - x)^2 - Product_{k>=1} (1 - x^(2^k)))/2. - Ilya Gutkovskiy, Apr 03 2019

A026465 Length of n-th run of identical symbols in the Thue-Morse sequence A010060 (or A001285).

Original entry on oeis.org

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

Offset: 1

Clark Kimberling

It appears that the sequence can be calculated by any of the following methods:
(1) Start with 1 and repeatedly replace 1 with 1, 2, 1 and 2 with 1, 2, 2, 2, 1;
(2) a(1) = 1, all terms are either 1 or 2 and, for n > 0, a(n) = 1 if the length of the n-th run of 2's is 1; a(n) = 2 if the length of the n-th run of consecutive 2's is 3, with each run of 2's separated by a run of two 1's;
(3) replace each 3 in A080426 with 2. - John W. Layman, Feb 18 2003
Number of representations of n as a sum of Jacobsthal numbers (1 is allowed twice as a part). Partial sums are A003159. With interpolated zeros, g.f. is (Product_{k>=1} (1 + x^A078008(k)))/2. - Paul Barry, Dec 09 2004
In other words, the consecutive 0's or 1's in A010060 or A010059. - Robin D. Saunders (saunders_robin_d(AT)hotmail.com), Sep 06 2006
From Carlo Carminati, Feb 25 2011: (Start)
The sequence (starting with the second term) can also be calculated by the following method:
Apply repeatedly to the string S_0 = [2] the following algorithm: take a string S, double it, if the last figure is 1, just add the last figure to the previous one, if the last figure is greater than one, decrease it by one unit and concatenate a figure 1 at the end. (This algorithm is connected with the interpretation of the sequence as a continued fraction expansion.) (End)
This sequence, starting with the second term, happens to be the continued fraction expansion of the biggest cluster point of the set {x in [0,1]: F^k(x) >= x, for all k in N}, where F denotes the Farey map (see A187061). - Carlo Carminati, Feb 28 2011
Starting with the second term, the fixed point of the substitution 2 -> 211, 1 -> 2. - Carlo Carminati, Mar 03 2011
It appears that this sequence contains infinitely many distinct palindromic subsequences. - Alexander R. Povolotsky, Oct 30 2016
From Michel Dekking, Feb 13 2019: (Start)
Let tau defined by tau(0) = 01, tau(1) = 10 be the Thue-Morse morphism, with fixed point A010060. Consecutive runs in A010060 are 0, 11, 0, 1, 00, 1, 1, ..., which are coded by their lengths 1, 2, 1, 1, 2, ... Under tau^2 consecutive runs are mapped to consecutive runs:
      tau^2(0) = 0110, tau^2(1) = 1001,
      tau^2(00) = 01100110, tau^2(11) = 10011001.
The reason is that (by definition of a run!) runs of 0's and runs of 1's alternate in the sequence of runs, and this is inherited by the image of these runs under tau^2.
Under tau^2 the runs of length 1 are mapped to the sequence 1,2,1 of run lengths, and the runs of length 2 are mapped to the sequence 1,2,2,2,1 of run lengths. This proves John Layman's conjecture number (1): it follows that (a(n)) is fixed point of the morphism alpha
      alpha: 1 -> 121, 2 -> 12221.
Since alpha(1) and alpha(2) are both palindromes, this also proves Alexander Povolotsky's conjecture.
(End)

N. J. A. Sloane, Table of n, a(n) for n = 1..10000
G. Allouche, Jean-Paul Allouche and Jeffrey Shallit, Kolam indiens, dessins sur le sable aux îles Vanuatu, courbe de Sierpinski et morphismes de monoïde, Ann. Inst. Fourier (Grenoble), Vol. 56, No. 7 (2006), pp. 2115-2130.
Jean-Paul Allouche, On the morphism 1 -> 121, 2 -> 12221, CNRS France, 2024. See pp. 1-3, 7.
Jean-Paul Allouche, On the morphism 1 -> 121, 2 -> 12221, Preprint, 2024 [Local copy, with permission]
Jean-Paul Allouche, Andre Arnold, Jean Berstel, Srecko Brlek, William Jockusch, Simon Plouffe and Bruce E. Sagan, A sequence related to that of Thue-Morse, Discrete Math., Vol. 139, No. 1-3 (1995), pp. 455-461.
Claudio Bonanno, Carlo Carminati, Stefano Isola and Giulio Tiozzo, Dynamics of continued fractions and kneading sequences of unimodal maps, arXiv:1012.2131 [math.DS], 2010-2012.
Srećko Brlek, Enumeration of factors in the Thue-Morse word, Discrete Applied Math., Vol. 24, No. 1-3 (1989), pp. 83-96.
Julien Cassaigne, Limit values of the recurrence quotient of Sturmian sequences, Theoret. Comput. Sci., Vol. 218, No. 1 (1999), pp. 3-12.
Artūras Dubickas, On the distance from a rational power to the nearest integer, Journal of Number Theory, Vol. 117, No. 1 (March 2006), pp. 222-239.
Artūras Dubickas, On a sequence related to that of Thue-Morse and its applications, Discrete Math., Vol. 307, No. 9-10 (2007), pp. 1082-1093. MR2292537 (2008b:11086).
Cor Kraaikamp, Thomas A. Schmidt and Wolfgang Steiner, Natural extensions and entropy of alpha-continued fractions, arXiv:1011.4283 [math.DS], 2010-2012.
Claude Lenormand, Deux transformations sur les mots, Preprint, 5 pages, Nov 17 2003. Apparently unpublished. This is a scanned copy of the version that the author sent to me in 2003. - _N. J. A. Sloane_, Sep 09 2018. See page 2.
Kevin Ryde, PARI/GP Code
Jeffrey Shallit, Automaticity IV: Sequences, sets, and diversity, J. Théor. Nombres Bordeaux, Vol. 8, No. 2 (1996), pp. 347-367. See page 354.

Cf. A010060, A001285, A101615, A026490 (run lengths).
A080426 is an essentially identical sequence with another set of constructions.
Cf. A104248 (bisection odious), A143331 (bisection evil), A003159 (partial sums).
Cf. A187061, A363361 (as continued fraction).

import Data.List (group)
a026465 n = a026465_list !! (n-1)
a026465_list = map length $ group a010060_list
-- Reinhard Zumkeller, Jul 15 2014

# From Carlo Carminati, Feb 25 2011:
## period-doubling routine:
double:=proc(SS)
NEW:=[op(S), op(S)]:
if op(nops(NEW),NEW)=1
then NEW:=[seq(op(j,NEW), j=1..nops(NEW)-2),op(nops(NEW)-1,NEW)+1]:
else NEW:=[seq(op(j,NEW), j=1..nops(NEW)-1),op(nops(NEW)-1,NEW)-1,1]:
fi:
end proc:
# 10 loops of the above routine generate the first 1365 terms of the sequence
# (except for the initial term):
S:=[2]:
for j from 1 to 10  do S:=double(S); od:
S;
# From N. J. A. Sloane, Dec 31 2013:
S:=[b]; M:=14;
for n from 1 to M do T:=subs({b=[b,a,a], a=[b]}, S);
    S := map(x->op(x),T); od:
T:=subs({a=1,b=2},S): T:=[1,op(T)]: [seq(T[n],n=1..40)];

Length /@ Split@ Nest[ Flatten@ Join[#, # /. {1 -> 2, 2 -> 1}] &, {1}, 7]
NestList[ Flatten[# /. {1 -> {2}, 2 -> {1, 1, 2}}] &, {1}, 7] // Flatten (* Robert G. Wilson v, May 20 2014 *)

PARI
```
\\ See links.
```

def A026465(n):
    if n==1: return 1
    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,n)-iterfun(lambda x:f(x)+n-1,n-1) # Chai Wah Wu, Jan 29 2025

a(1) = 1; for n > 1, a(n) = A003159(n) - A003159(n-1). - Benoit Cloitre, May 31 2003
G.f.: Product_{k>=1} (1 + x^A001045(k)). - Paul Barry, Dec 09 2004
Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 3/2. - Amiram Eldar, Jan 16 2022

Corrected and extended by John W. Layman, Feb 18 2003

Definition revised by N. J. A. Sloane, Dec 30 2013

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

A026147 a(n) = position of n-th 1 in A001285 or A010059 (Thue-Morse sequence).

Original entry on oeis.org

1, 4, 6, 7, 10, 11, 13, 16, 18, 19, 21, 24, 25, 28, 30, 31, 34, 35, 37, 40, 41, 44, 46, 47, 49, 52, 54, 55, 58, 59, 61, 64, 66, 67, 69, 72, 73, 76, 78, 79, 81, 84, 86, 87, 90, 91, 93, 96, 97, 100, 102, 103, 106, 107, 109, 112, 114, 115, 117, 120, 121, 124, 126, 127, 130

Offset: 1

Clark Kimberling

Barbeau notes that if we let A = the first 2^k terms of this sequence and B = the first 2^k terms of A181155, then the two sets A and B 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

Edward J. Barbeau, Power Play, MAA, 1997. See p. 104.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A001969, A181155.

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

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

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

def A026147(n): return 1+((m:=n-1).bit_count()&1)+(m<<1) # Chai Wah Wu, Mar 03 2023

a(n) = 1+A001969(n).
a(n) = Sum_{k=0..2n} mod(-2 + Sum_{j=0..k} floor(C(k, j)/2), 3). - Paul Barry, Dec 24 2004
a(n) + A010059(n+1) = 2n + 2 for n >= 0. - Clark Kimberling, Oct 06 2014

A029886 Convolution of Thue-Morse sequence A001285 with itself.

Original entry on oeis.org

1, 4, 8, 10, 12, 14, 15, 16, 22, 24, 23, 26, 29, 30, 34, 40, 38, 40, 43, 42, 47, 50, 52, 56, 55, 56, 62, 66, 64, 70, 71, 64, 78, 80, 75, 82, 83, 82, 88, 96, 89, 92, 100, 98, 102, 106, 105, 104, 111, 112, 114, 122, 118, 122, 125, 120, 130, 136, 131, 130, 141, 134, 138, 160

Offset: 0

N. J. A. Sloane

nonn

Comment from Jeremy Gardiner, Dec 28 2008: The following sequences all appear to have the same parity: A003071, A029886, A061297, A092524, A093431, A102393, A104258, A122248, A128975.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Tanya Khovanova, There are no coincidences, arXiv preprint 1410.2193 [math.CO], 2014.

Cf. A001285.

P[n_, x_] := (bb = IntegerDigits[n, 2]) . x^Range[Length[bb]-1, 0, -1];
TM[n_] := 1 + Mod[P[n, 1], 2];
a[n_] := Sum[TM[k] TM[n-k], {k, 0, n}];
Table[a[n], {n, 0, 63}] (* Jean-François Alcover, Aug 31 2018 *)

a(n)=sum(k=0,n,(1+subst(Pol(binary(k)),x,1)%2)*(1+subst(Pol(binary(n-k)),x,1)%2)) \\ Ralf Stephan, Aug 23 2013

G.f.: (1/4)*(3/(1 - x) - Product_{k>=0} (1 - x^(2^k)))^2. - Ilya Gutkovskiy, Apr 03 2019

A026492 a(n) = t(3n), where t = A001285 (Thue-Morse sequence).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Michael Gilleland, Some Self-Similar Integer Sequences

Cf. A001285, A026498, A026513.

Array[1 + Mod[DigitCount[3 #, 2, 1], 2] &, 105, 0] (* Michael De Vlieger, Oct 03 2019 *)

a(n)=1+hammingweight(3*n)%2 \\ Charles R Greathouse IV, May 09 2016

Name and Pari adapted to match offset in A001285 by Sean A. Irvine, Oct 02 2019

A029885 Boustrophedon transform of 1 followed by Thue-Morse sequence A001285.

Original entry on oeis.org

1, 2, 5, 13, 34, 108, 415, 1841, 9381, 53733, 342086, 2395481, 18300250, 151453434, 1349856656, 12890177378, 131298281746, 1420980348324, 16283235530691, 196958363484995, 2507751773736087, 33526171616091612

Offset: 0

N. J. A. Sloane

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 0..400
Peter Luschny, An old operation on sequences: the Seidel transform.
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory, 17A (1996) 44-54 (Abstract, pdf, ps).
N. J. A. Sloane, Transforms.
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform

Cf. A109449, A001285, A230958, A230950, A230951.

a029885 n = sum $ zipWith (*) (a109449_row n) (1 : map fromIntegral a001285_list)
-- Reinhard Zumkeller, Nov 04 2013

tm[n_] := Mod[Sum[Mod[Binomial[n, k], 2], {k, 0, n}], 3];
T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}];
a[n_] := Sum[T[n, k] If[k == 0, 1, tm[k - 1]], {k, 0, n}];
Table[a[n], {n, 0, 21}] (* Jean-François Alcover, Jul 02 2019 *)

Definition corrected by Reinhard Zumkeller, Nov 04 2013

A026498 a(n) = t(1+3n), where t = A001285 (Thue-Morse sequence).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..10000

Cf. A001285, A026492, A026513.

Array[1 + Mod[DigitCount[3 # + 1, 2, 1], 2] &, 105, 0] (* Michael De Vlieger, Oct 03 2019 *)

a(n)=1+hammingweight(3*n+1)%2 \\ Charles R Greathouse IV, May 09 2016

Name and Pari adapted to match offset in A001285 by Sean A. Irvine, Oct 03 2019

A026513 a(n) = t(2+3n), where t = A001285 (Thue-Morse sequence).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

nonn

Michael De Vlieger, Table of n, a(n) for n = 0..9999

Cf. A001285, A026492, A026498.

Array[1 + Mod[DigitCount[3 # + 2, 2, 1], 2] &, 105, 0] (* Michael De Vlieger, Oct 06 2019 *)
Table[ThueMorse[2+3n],{n,0,100}]+1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Nov 29 2020 *)

a(n)=1+hammingweight(3*n+2)%2 \\ Charles R Greathouse IV, May 09 2016

Name and Pari adapted to match offset in A001285 by Sean A. Irvine, Oct 05 2019

A080814 Successive words in the formal D0L language that produces the Thue-Morse sequence A001285 (start with 1, map 1 -> 12, 2 -> 21).

Original entry on oeis.org

1, 12, 1221, 12212112, 1221211221121221, 12212112211212212112122112212112, 1221211221121221211212211221211221121221122121121221211221121221

Offset: 1

N. J. A. Sloane, Mar 26 2003

nonn

A. Salomaa, Jewels of Formal Language Theory. Computer Science Press, Rockville, MD, 1981, p. 5.

Jarosław Grytczuk, Thue type problems for graphs, points and numbers, Discrete Math., 308 (2008), 4419-4429.

Cf. A001285 (which is the limiting word), A010060, A080815.

Map[FromDigits,SubstitutionSystem[{1->{1,2},2->{2,1}},{1},7]] (* Paolo Xausa, Dec 24 2023 *)

cp's OEIS Frontend

A026430 a(n) is the sum of first n terms of A001285 (Thue-Morse sequence).

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A026465 Length of n-th run of identical symbols in the Thue-Morse sequence A010060 (or A001285).

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A029883 First differences of Thue-Morse sequence A001285.

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A026147 a(n) = position of n-th 1 in A001285 or A010059 (Thue-Morse sequence).

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A029886 Convolution of Thue-Morse sequence A001285 with itself.

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A026492 a(n) = t(3n), where t = A001285 (Thue-Morse sequence).

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A029885 Boustrophedon transform of 1 followed by Thue-Morse sequence A001285.