A001285 -id:A001285 - cp's OEIS frontend

A026491 a(n) = least k > a(n-1) such that A001285(k-1) = A001285(n-1) for n >= 1.

1, 4, 5, 8, 10, 12, 13, 16, 17, 20, 21, 24, 26, 28, 29, 32, 34, 36, 37, 40, 42, 44, 45, 48, 49, 52, 53, 56, 58, 60, 61, 64, 65, 68, 69, 72, 74, 76, 77, 80, 81, 84, 85, 88, 90, 92, 93, 96, 98, 100, 101, 104, 106, 108, 109, 112, 113, 116, 117

Offset: 0

Clark Kimberling

nonn

Robert Israel, Table of n, a(n) for n = 0..10000

Cf. A001285, A171900.

N:= 200: # for terms <= N
S:= StringTools:-ThueMorse(N):
R:= 1: r:= 1:
for n from 1 do
  j:= SearchText(S[n],S,r+1..-1);
  if j = 0 then break fi;
  R:= R, r+j;
  r:= r+j;
  if r >= N then break fi;
od:
R; # Robert Israel, Apr 11 2019
# alternative
A026491 := proc(n)
    option remember ;
    local f,k ;
    if n = 0 then
        1;
    else
        f := A001285(n-1) ;
        for k from procname(n-1)+1 do
            if A001285(k-1) = f then
                return k;
            end if;
        end do:
    end if;
end proc:
seq(A026491(n),n=0..40) ; # R. J. Mathar, Jun 24 2021

a[n_] := a[n] = If[n==0, 1, For[k = a[n-1]+1, True, k++, If[ThueMorse[k-1]==ThueMorse[n-1], Return[k]]]];
Table[a[n], {n, 0, 60}] (* Jean-François Alcover, Sep 16 2022 *)

a(n+1)-a(n) = A171900(n+2) for n>=1. - Michel Dekking, Apr 11 2019

Clarified NAME with respect to A001285 offsets. - R. J. Mathar, Jun 24 2021

A026517 a(n) = t(5n), where t = A001285 (Thue-Morse sequence).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

nonn

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

Cf. A001285.

ThueMorse[Range[0, 500, 5]] + 1 (* Paolo Xausa, Nov 28 2024 *)

a(n)=1+hammingweight(5*n)%2 \\ Charles R Greathouse IV, May 09 2016

Name and Pari code corrected by Sean A. Irvine, Oct 05 2019

A080815 Successive strings in the construction of the Thue-Morse sequence A001285.

Original entry on oeis.org

1, 2, 21, 2112, 21121221, 2112122112212112, 21121221122121121221211221121221, 2112122112212112122121122112122112212112211212212112122112212112

Offset: 1

N. J. A. Sloane, Mar 26 2003

nonn

A. Salomaa, Jewels of Formal Language Theory. Computer Science Press, Rockville, MD, 1981, p. 6.

Cf. A080814.

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

Drop first half of each word in A080814.

a(8) from Paolo Xausa, Dec 24 2023

A225186 Squares in the Thue-Morse word A001285.

Original entry on oeis.org

11, 22, 1212, 2121, 121121, 212212, 12211221, 21122112, 122112122112, 211221211221, 1221211212212112, 2112122121121221, 122121121221122121121221, 211212212112211212212112, 12212112211212211221211221121221, 21121221122121122112122112212112

Offset: 1

N. J. A. Sloane, May 04 2013

nonn

A square in an infinite word w is a factor of the form uu, where u has finite length.
Consists of 11, 22, 121121, 212212 and their repeated images under the morphism 1 -> 12, 2 -> 21.
Replacing {1,2} by {0,1} gives the squares in A010060.

James D. Currie and Narad Rampersad, Infinite words containing squares at every position, arXiv:0803.1189 [math.CO], 2008; or also, RAIRO-Theor. Inf. Appl. 44, 113-124 (2010).

Cf. A001285, A020060, A225188 (square roots).

A230958 Boustrophedon transform of Thue-Morse sequence A001285.

Original entry on oeis.org

1, 3, 7, 15, 39, 127, 480, 2143, 10907, 62495, 397814, 2785861, 21282228, 176133285, 1569817724, 14990658724, 152693582275, 1652531857935, 18936620009722, 229053108410969, 2916394751599614, 38989325834726043, 546070266163669664, 7995699956778626764

Offset: 0

Reinhard Zumkeller, Nov 04 2013

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 Ser. A, 76(1) (1996), 44-54 (Abstract, pdf, ps).
J. Millar, N. J. A. Sloane and N. E. Young, A new operation on sequences: the Boustrophedon transform, J. Combin. Theory Ser. A, 76(1) (1996), 44-54.
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [USA access only through the HATHI TRUST Digital Library]
Ludwig Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187. [Access through ZOBODAT]
Wikipedia, Boustrophedon transform.
Index entries for sequences related to boustrophedon transform

Cf. A029885, A230950, A230951.

Cf. A001285, A109449.

a230958 n = sum $ zipWith (*) (a109449_row n) $ map fromIntegral a001285_list

T[n_, k_] := (n!/k!) SeriesCoefficient[(1 + Sin[x])/Cos[x], {x, 0, n - k}];
tm[n_] := Mod[Sum[Mod[Binomial[n, k], 2], {k, 0, n}], 3];
Table[Sum[T[n, k] tm[k], {k, 0, n}], {n, 0, 23}] (* Jean-François Alcover, Jul 23 2019 *)

from itertools import accumulate, count, islice
def A230958_gen(): # generator of terms
    blist = tuple()
    for i in count(0):
        yield (blist := tuple(accumulate(reversed(blist), initial=2 if i.bit_count()&1 else 1)))[-1]
A230958_list = list(islice(A230958_gen(),30)) # Chai Wah Wu, Apr 17 2023

a(n) = Sum_{k=0..n} A109449(n,k)*A001285(k).

A029878 Inverse Euler transform of {A001285(0), A001285(1), ...} where A001285(n) is Thue-Morse sequence.

Original entry on oeis.org

1, 1, 0, -2, 1, 0, -1, 1, 2, -3, -2, 5, 1, -6, 1, 8, -6, -10, 14, 7, -25, 8, 36, -34, -41, 72, 25, -125, 29, 187, -150, -216, 361, 137, -657, 159, 977, -810, -1135, 1937, 752, -3558, 792, 5361, -4327, -6318, 10641, 4281, -19848, 4286

Offset: 1

N. J. A. Sloane

sign

			(1-x)^(-1)*(1-x^2)^(-1)*(1-x^4)^2*(1-x^5)^(-1)* ... = 1 + x + 2*x^2 + 2*x^3 + x^4 + 2*x^5 + ... .

Seiichi Manyama, Table of n, a(n) for n = 1..5000
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Euler Transform

Cf. A001285, A316149.

Product_{k>=1} (1-x^k)^(-a(k)) = 1 + Sum_{k>=1} A001285(k-1)*x^k. - Seiichi Manyama, Jun 25 2018

Name edited by Seiichi Manyama, Jun 25 2018

A214215 List of subwords (or factors) of the Thue-Morse "1,2"-word A001285.

Original entry on oeis.org

1, 2, 11, 12, 21, 22, 112, 121, 122, 211, 212, 221, 1121, 1122, 1211, 1212, 1221, 2112, 2121, 2122, 2211, 2212, 11212, 11221, 12112, 12122, 12211, 12212, 21121, 21122, 21211, 21221, 22112, 22121, 112122, 112211, 112212, 121121, 121122, 121221, 122112, 122121

Offset: 1

N. J. A. Sloane, Jul 10 2012

nonn

The number of factors of length m is given by A005942(m).

Alois P. Heinz, Table of n, a(n) for n = 1..10200

Cf. A001285, A010060, A005942.

b:= proc(n) option remember; local r;
      `if`(n=0, 1, `if`(n<4, 2*n, `if`(irem(n, 2, 'r')=0,
       b(r)+b(r+1), 2*b(r+1))))
    end:
m:= proc(n) option remember; local r;
      `if`(n=0, 1, `if`(irem(n, 2, 'r')=0, m(r), 3-m(r)))
    end:
T:= proc(n) local k, s; s:={};
      for k while nops(s)Alois P. Heinz, Jul 19 2012

b[n_] := b[n] = Module[{r}, If[n == 0, 1, If[n < 4, 2n, r = Quotient[n, 2]; If[Mod[n, 2] == 0, b[r] + b[r + 1], 2b[r + 1]]]]];
m[n_] := m[n] = Module[{r}, If[n == 0, 1, r = Quotient[n, 2]; If[Mod[n, 2] == 0, m[r], 3 - m[r]]]];
T[n_] := Module[{k, s = {}}, For[k = 1,  Length[s] < b[n], k++, s = s  ~Union~ {FromDigits[#]}& @ Table[m[i], {i, k, k + n - 1}]]; Sort[s]];
Array[T, 10] // Flatten (* Jean-François Alcover, Nov 22 2020, after Alois P. Heinz *)

A316149 Inverse Euler transform of Thue-Morse sequence A001285.

Original entry on oeis.org

2, -1, -1, 2, -3, 3, 0, -4, 6, -6, 6, -1, -12, 24, -29, 23, 9, -64, 114, -132, 81, 78, -333, 577, -627, 279, 610, -1896, 2979, -2911, 672, 4232, -10754, 15576, -13515, -591, 28098, -61548, 81664, -60408, -27030, 180784, -351081, 425892, -253838, -281760, 1140396, -1995767, 2195952, -930876

Offset: 1

Seiichi Manyama, Jun 25 2018

sign

			(1-x)^(-2)*(1-x^2)*(1-x^3)*(1-x^4)^(-2)* ... = 1 + 2*x + 2*x^2 + x^3 + 2*x^4 + ... .

Seiichi Manyama, Table of n, a(n) for n = 1..5000
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Euler Tranceform

Cf. A001285, A029878.

# The function EulerInvTransform is defined in A358451.
a := EulerInvTransform(A001285):
seq(a(n), n = 1..50); # Peter Luschny, Nov 21 2022

Product_{k>=1} (1-x^k)^(-a(k)) = 1 + Sum_{k>=1} A001285(k)*x^k.

A029881 Möbius transform of Thue-Morse sequence A001285 (when shifted once right).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000
N. J. A. Sloane, Transforms
Index entries for sequences related to binary expansion of n

Cf. A001285, A008683.

A029881(n) = sumdiv(n,d,moebius(n/d)*(1+(hammingweight(d-1)%2))); \\ Antti Karttunen, Dec 15 2024

a(n) = Sum_{d|n} A008683(n/d)*A001285(d-1). - Antti Karttunen, Dec 15 2024

Starting offset corrected (from 0 to 1), more terms added and definition clarified by Antti Karttunen, Dec 15 2024

A029884 Second differences of Thue-Morse sequence A001285.

Original entry on oeis.org

-1, -1, 2, -2, 1, 1, -1, -1, 1, 1, -2, 2, -1, -1, 2, -2, 1, 1, -2, 2, -1, -1, 1, 1, -1, -1, 2, -2, 1, 1, -1, -1, 1, 1, -2, 2, -1, -1, 1, 1, -1, -1, 2, -2, 1, 1, -2, 2, -1, -1, 2, -2, 1, 1, -1, -1, 1, 1, -2, 2, -1, -1, 2, -2, 1, 1, -2, 2, -1, -1, 1, 1, -1, -1, 2, -2, 1, 1, -2, 2, -1

Offset: 0

N. J. A. Sloane

sign

cp's OEIS Frontend

A026491 a(n) = least k > a(n-1) such that A001285(k-1) = A001285(n-1) for n >= 1.

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A026517 a(n) = t(5n), where t = A001285 (Thue-Morse sequence).

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A080815 Successive strings in the construction of the Thue-Morse sequence A001285.

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A225186 Squares in the Thue-Morse word A001285.

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A230958 Boustrophedon transform of Thue-Morse sequence A001285.

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A029878 Inverse Euler transform of {A001285(0), A001285(1), ...} where A001285(n) is Thue-Morse sequence.

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A214215 List of subwords (or factors) of the Thue-Morse "1,2"-word A001285.

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Author

Keywords

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Maple

Mathematica

A316149 Inverse Euler transform of Thue-Morse sequence A001285.

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Programs

Maple

Formula

A029881 Möbius transform of Thue-Morse sequence A001285 (when shifted once right).