A001285 -id:A001285 - cp's OEIS frontend

A080826 Successive factors of the Thue-Morse word defined by sequence A001285.

122, 12, 1122, 11212212, 1121221122121122, 11212211221211212212112211212212, 1121221122121121221211221121221122121122112122121121221122121122

Offset: 1

N. J. A. Sloane, Mar 28 2003

The next term is too large to include.

G. Melançon, Home page [Dead link]
G. Melançon, Lyndon Factorization of Infinite Words STACS 96 (Grenoble, 1996), 147-154, Lecture Notes in Comput. Sci., 1046, Springer, Berlin, 1996.

A080826:= proc(n) option remember: local l,i: if n = 0 then RETURN([1,2,2]) else l:=A080826(n-1):l := subsop(nops(l)=NULL,l):RETURN([op(l),seq(op(A080826(i)),i=0..n-2)]) fi:end; ds:=s->add(s[j]*10^(nops(s)-j),j=1..nops(s)); seq(ds(A080826(n)),n=0..7); (C. Ronaldo)

More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Dec 19 2004

A085297 Nonzero residues of Catalan sequence modulo 3; related to the Thue-Morse sequence (A001285).

Original entry on oeis.org

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

Offset: 1

Paul D. Hanna, Jun 24 2003

nonn

The runs of zeros in between the digit strings are given in A085296.

Cf. A001285 (Thue-Morse), A000108 (Catalan), A039969 (d-perfect), A085296.

If a leading '1' is added to the Catalan sequence modulo 3, the only nonzero digit strings are {1, 1, 1, 2, 2, 2} and {2, 2, 2, 1, 1, 1}. Replacing these digit strings with their first digit, {1, 1, 1, 2, 2, 2} -> '1' and {2, 2, 2, 1, 1, 1} -> '2', then omitting all zeros, results in the Thue-Morse sequence.

A225188 Words u such that uu is a factor of the Thue-Morse word A001285.

Original entry on oeis.org

1, 2, 12, 21, 121, 212, 1221, 2112, 122112, 211221, 12212112, 21121221, 122121121221, 211212212112, 1221211221121221, 2112122112212112, 122121122112122112212112, 211212211221211221121221, 12212112211212212112122112212112, 21121221122121121221211221121221, 122121122112122121121221122121121221211221121221, 211212211221211212212112211212212112122112212112

Offset: 1

N. J. A. Sloane, May 05 2013

nonn

Consists of 1, 2, 121, 212 and their repeated images under the morphism 1 -> 12, 2 -> 21.
These are the square roots of the words in A225186.
Replacing {1,2} by {0,1} gives the words u such that uu is a factor of 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, A225186.

A026535 a(n) = t(1+5n) where t = A001285 (Thue-Morse sequence).

Original entry on oeis.org

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

Offset: 0

Clark Kimberling

nonn

Cf. A001285, A026517.

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

Name and Pari corrected by Sean A. Irvine, Oct 06 2019

A029877 Euler transform of Thue-Morse sequence A001285.

Original entry on oeis.org

1, 3, 5, 9, 15, 25, 38, 61, 91, 137, 200, 293, 416, 595, 832, 1161, 1601, 2198, 2983, 4042, 5419, 7245, 9615, 12714, 16703, 21879, 28492, 36989, 47795, 61561, 78956, 100986, 128653, 163461, 206981, 261393

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane, Transforms

A029879 Binomial transform of Thue-Morse sequence A001285.

Original entry on oeis.org

1, 3, 7, 14, 27, 52, 100, 192, 371, 728, 1454, 2940, 5968, 12080, 24304, 48608, 96851, 192816, 384562, 769636, 1546226, 3116184, 6290984, 12700832, 25603536, 51479808, 103178992, 206133344, 410674512, 816542272, 1621848640, 3221225472, 6403020371, 12746005088

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane, Transforms

Cf. A001285.

A029880 Inverse binomial transform of Thue-Morse sequence A001285.

Original entry on oeis.org

1, 1, -1, 0, 3, -10, 24, -48, 83, -126, 170, -208, 240, -272, 272, 0, -1453, 6698, -22050, 60896, -148430, 326404, -653280, 1188640, -1939568, 2730976, -2946256, 1085760, 5849808, -22935520, 57106944, -114213888, 188997203

Offset: 0

N. J. A. Sloane

sign

N. J. A. Sloane, Transforms

Join[{1}, Table[Differences[Table[ThueMorse[n], {n, 0, m}], m][[1]], {m, 1, 10}]] (* Robert P. P. McKone, Dec 18 2020 *)

A029882 Inverse Moebius transform of Thue-Morse sequence A001285.

Original entry on oeis.org

1, 3, 3, 4, 3, 6, 2, 6, 5, 6, 2, 9, 2, 6, 7, 7, 3, 9, 2, 9, 5, 6, 3, 12, 4, 6, 7, 8, 3, 12, 2, 9, 6, 6, 5, 14, 2, 6, 6, 12, 2, 12, 3, 8, 11, 6, 2, 15, 3, 9, 7, 8, 3, 12, 5, 12, 6, 6, 2, 19, 2, 6, 9, 10, 6, 12, 2, 9, 6, 12, 3, 18, 2, 6, 10

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane, Transforms

A029888 Convolution of Thue-Morse sequence A001285 with primes.

Original entry on oeis.org

2, 7, 15, 25, 42, 62, 87, 116, 152, 192, 241, 296, 356, 425, 491, 569, 660, 754, 851, 960, 1072, 1179, 1311, 1447, 1584, 1745, 1899, 2055, 2220, 2388, 2573, 2778, 2992, 3188, 3415, 3640, 3868, 4125, 4383

Offset: 0

N. J. A. Sloane

nonn

tm[0] = 1; tm[n_?EvenQ] := tm[n] = tm[n/2]; tm[n_] := tm[n] = 3 - tm[(n-1)/2]; a[n_] := ListConvolve[tm /@ Range[0, n], Prime /@ Range[n+1]] // First; Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Nov 18 2013 *)

A029896 Convolution of Thue-Morse sequence A001285 with A008578 = {1, primes}.

Original entry on oeis.org

1, 4, 9, 16, 27, 43, 63, 89, 118, 153, 193, 243, 297, 358, 427, 492, 571, 661, 755, 853, 961, 1074, 1181, 1312, 1448, 1586, 1747, 1900, 2057, 2221, 2389, 2575, 2780, 2993, 3189, 3417, 3641, 3870, 4127, 4384

Offset: 0

N. J. A. Sloane

nonn

cp's OEIS Frontend

A080826 Successive factors of the Thue-Morse word defined by sequence A001285.

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A085297 Nonzero residues of Catalan sequence modulo 3; related to the Thue-Morse sequence (A001285).

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Formula

A225188 Words u such that uu is a factor of the Thue-Morse word A001285.

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A026535 a(n) = t(1+5n) where t = A001285 (Thue-Morse sequence).

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PARI

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A029877 Euler transform of Thue-Morse sequence A001285.

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A029879 Binomial transform of Thue-Morse sequence A001285.

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A029880 Inverse binomial transform of Thue-Morse sequence A001285.

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Mathematica

A029882 Inverse Moebius transform of Thue-Morse sequence A001285.

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A029888 Convolution of Thue-Morse sequence A001285 with primes.

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Programs

Mathematica

A029896 Convolution of Thue-Morse sequence A001285 with A008578 = {1, primes}.

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