A074650 -id:A074650 - cp's OEIS frontend

A060221 Number of orbits of length n under the full 18-shift (whose periodic points are counted by A001027).

18, 153, 1938, 26163, 377910, 5667681, 87460002, 1377481950, 22039920504, 357046533675, 5842582734474, 96402612275775, 1601766528128550, 26772383354990049, 449776041098370870, 7589970692848393200, 128583032925805678350, 2185911559727674682148, 37275544492386193492506

Offset: 1

Thomas Ward, Mar 21 2001

Number of Lyndon words (aperiodic necklaces) with n beads of 18 colors. - Andrew Howroyd, Dec 10 2017

			a(2)=153 since there are 324 points of period 2 in the full 18-shift and 18 fixed points, so there must be (324-18)/2 = 153 orbits of length 2.

G. C. Greubel, Table of n, a(n) for n = 1..792
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
T. Ward, Exactly realizable sequences

Column 18 of A074650.

Cf. A001027, A008683.

A060221:= func< n | (1/n)*(&+[MoebiusMu(d)*(18)^Floor(n/d): d in Divisors(n)]) >;
[A060221(n): n in [1..40]]; // G. C. Greubel, Sep 13 2024

A060221[n_]:= DivisorSum[n, (18)^(n/#)*MoebiusMu[#] &]/n;
Table[A060221[n], {n, 40}] (* G. C. Greubel, Sep 13 2024 *)

a001027(n) = 18^n;
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001027(n/d)); \\ Michel Marcus, Sep 11 2017

def A060221(n): return (1/n)*sum(moebius(k)*(18)^(n/k) for k in (1..n) if (k).divides(n))
[A060221(n) for n in range(1,41)] # G. C. Greubel, Sep 13 2024

a(n) = (1/n)* Sum_{d|n} mu(d)*A001027(n/d).

G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 18*x^k))/k. - Ilya Gutkovskiy, May 20 2019

More terms from Michel Marcus, Sep 11 2017

A060222 Number of orbits of length n under the full 19-shift (whose periodic points are counted by A001029).

Original entry on oeis.org

19, 171, 2280, 32490, 495216, 7839780, 127695960, 2122929090, 35854187880, 613106378136, 10590023536200, 184442905990860, 3234844881712080, 57071906063500860, 1012075135324821024, 18027588346914850290, 322375697516753069760, 5784852794310472599780, 104127350297911241532840

Offset: 1

Thomas Ward, Mar 21 2001

Number of monic irreducible polynomials of degree n over GF(19). - Andrew Howroyd, Dec 10 2017

			a(2)=171 since there are 361 points of period 2 in the full 19-shift and 19 fixed points, so there must be (361-19)/2 = 171 orbits of length 2.

Vincenzo Librandi, Table of n, a(n) for n = 1..799
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
T. Ward, Exactly realizable sequences

Column 19 of A074650.

Cf. A001029.

A060222:= func< n | (1/n)*(&+[MoebiusMu(d)*(19)^Floor(n/d): d in Divisors(n)]) >;
[A060222(n): n in [1..40]]; // G. C. Greubel, Sep 23 2024

a[n_]:=(1/n) Sum[MoebiusMu[d] 19^(n/d), {d, Divisors[n]}]; Table[a[n], {n, 20}] (* Vincenzo Librandi, Sep 19 2017 *)

a001029(n) = 19^n;
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001029(n/d)); \\ Michel Marcus, Sep 11 2017

def A060222(n): return (1/n)*sum(moebius(k)*(19)^(n/k) for k in (1..n) if (k).divides(n))
[A060222(n) for n in range(1, 41)] # G. C. Greubel, Sep 23 2024

a(n) = (1/n)* Sum_{d|n} mu(d)*A001029(n/d).

G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 19*x^k))/k. - Ilya Gutkovskiy, May 20 2019

More terms from Michel Marcus, Sep 11 2017

A292350 Number of Lyndon words (aperiodic necklaces) with 6 beads of n colors.

Original entry on oeis.org

0, 9, 116, 670, 2580, 7735, 19544, 43596, 88440, 166485, 295020, 497354, 804076, 1254435, 1897840, 2795480, 4022064, 5667681, 7839780, 10665270, 14292740, 18894799, 24670536, 31848100, 40687400, 51482925, 64566684, 80311266, 99133020, 121495355, 147912160

Offset: 1

Eric M. Schmidt, Dec 08 2017

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (7, -21, 35, -35, 21, -7, 1).

Row n=6 of A074650.

concat(0, Vec(x^2*(9 + 53*x + 47*x^2 + 11*x^3) / (1 - x)^7 + O(x^40))) \\ Colin Barker, Dec 08 2017

a(n) = (n^6 - n^3 - n^2 + n)/6.
From Colin Barker, Dec 08 2017: (Start)
G.f.: x^2*(9 + 53*x + 47*x^2 + 11*x^3) / (1 - x)^7.
a(n) = 7*a(n-1) - 21*a(n-2) + 35*a(n-3) - 35*a(n-4) + 21*a(n-5) - 7*a(n-6) + a(n-7) for n>7.
(End)

A352745 a(n) is the number of Lyndon factors of the Fibonacci string of length n.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 6, 5, 8, 6, 10, 7, 12, 8, 14, 9, 16, 10, 18, 11, 20, 12, 22, 13, 24, 14, 26, 15, 28, 16, 30, 17, 32, 18, 34, 19, 36, 20, 38, 21, 40, 22, 42, 23, 44

Offset: 0

Peter Luschny, Apr 06 2022

The Fibonacci string of length n is defined Fibonacci(n) = cat(Fibonacci(n - 1), Fibonacci(n - 2)) for 1 < n and the initial conditions Fibonacci(0) = "1" and Fibonacci(1) = "0", where 'cat' is the operation of concatenating strings. The length of Fibonacci(n) is A352744(1, n). The sequence starts: "1", "0", "01", "010", "01001", "01001010", ...

Apart from the first four terms seems to be identical with A117248.

			The Lyndon factorization of the Fibonacci strings of length n = 0..9.
[0] ["1"]
[1] ["0"]
[2] ["01"]
[3] ["01", "0"]
[4] ["01", "001"]
[5] ["01", "00101", "0"]
[6] ["01", "00101", "001", "001"]
[7] ["01", "00101", "0010010100101", "0"]
[8] ["01", "00101", "0010010100101", "00100101", "001", "001"]
[9] ["01", "00101", "0010010100101", "0010010100100101001010010010100101", "0"]

Guy Melançon, Lyndon factorization of infinite words, STACS 96 (Grenoble, 1996), 147-154, Lecture Notes in Comput. Sci., 1046, Springer, Berlin, 1996.
Wikipedia, Lyndon word

Cf. A000045, A014707, A074650, A117248, A211100, A352746, A352744.

with(StringTools): A352745 := n -> nops(LyndonFactors(Fibonacci(n))):
seq(A352745(n), n = 0..12);

A352746 a(n) is the number of Lyndon factors of the Thue-Morse string of length n.

Original entry on oeis.org

0, 1, 1, 1, 2, 2, 3, 4, 3, 3, 4, 5, 4, 5, 4, 4, 5, 4, 5, 6, 5, 6, 5, 5, 6, 7, 6, 5, 6, 5, 6, 7, 6, 5, 6, 7, 6, 7, 6, 6, 7, 8, 7, 6, 7, 6, 7, 8, 7, 8, 7, 7, 8, 6, 7, 8, 7, 6, 7, 8, 7, 8, 7, 7, 8, 6, 7, 8, 7, 8, 7, 7, 8, 9, 8, 7, 8, 7, 8, 9, 8, 9, 8, 8, 9, 7, 8

Offset: 0

Peter Luschny, Apr 06 2022

nonn

The Thue-Morse string of length n is the length-n prefix of the infinite Thue-Morse string. The sequence starts: "", "0", "01", "011", "0110", "01101", "011010", ...

			The Lyndon factorization of the Thue-Morse strings of length n = 0..9.
[0] []
[1] ["0"]
[2] ["01"]
[3] ["011"]
[4] ["011", "0"]
[5] ["011", "01"]
[6] ["011", "01", "0"]
[7] ["011", "01", "0", "0"]
[8] ["011", "01", "001"]
[9] ["011", "01", "0011"]

Augustin Ido and Guy Melançon, Lyndon factorization of the Thue-Morse word and its relatives, Discret. Math. Theor. Comput. Sci. 1997.
Wikipedia, Lyndon word

Cf. A000045, A014707, A074650, A211100, A352745.

with(StringTools): A352746 := n -> nops(LyndonFactors(ThueMorse(n))):
seq(A352746(n), n = 0..12);

A006174 Witt vector *3!.

Original entry on oeis.org

6, 27, 488, 7974, 149796, 2725447, 56970432, 1151053821, 25279412332, 543871341927, 12411512060544, 278163517356594, 6498314231705568, 149846653983570795, 3565206002960088128, 84045618111578025105

Offset: 1

Simon Plouffe

nonn

If c is the Witt transform of b then b(n) = Sum_{d|n} A074650(n/d, c(d)).

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

H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-108.
H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-107. (Annotated scanned copy)

Witt transform of A074651.

More terms and formula from Christian G. Bower, Aug 28 2002

A006175 Witt vector *4!.

Original entry on oeis.org

24, 972, 118592, 15210414, 2344956480, 377420590432, 67501965869568, 12329221295657241, 2383082885396731968, 467786496795764717088, 95188347941581635319296, 19578329367376510676884584

Offset: 1

Simon Plouffe

nonn

If c is the Witt transform of b then b(n) = Sum_{d|n} A074650(n/d, c(d)).

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

H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-108.
H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-107. (Annotated scanned copy)

Witt transform of A074652.

More terms and formula from Christian G. Bower, Aug 28 2002

A006176 Witt vector *5!.

Original entry on oeis.org

120, 49500, 55480000, 75108093750, 124667171985024, 226899085942554400, 453922674315047424000, 954267187464733528198125, 2112236210012497151219800000, 4825706564405954731805322783552

Offset: 1

Simon Plouffe

nonn

If c is the Witt transform of b then b(n) = Sum_{d|n} A074650(n/d, c(d)).

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

H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-108.
H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-107. (Annotated scanned copy)

Witt transform of A074653.

More terms and formula from Christian G. Bower, Aug 28 2002

A006178 Witt vector *3!/3!.

Original entry on oeis.org

1, 7, 93, 1419, 25225, 472037, 9501737, 196190781, 4219610242, 92198459515, 2068590840349, 46897782768404, 1083052539395723, 25199771186287195, 594383312662808405, 14098935496013599680, 337939791145403719897

Offset: 1

Simon Plouffe

nonn

If c is the Witt transform of b then b(n) = Sum_{d|n} A074650(n/d, c(d)).

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

H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-108.
H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-107. (Annotated scanned copy)

Witt transform of A029808.

More terms and formula from Christian G. Bower, Aug 28 2002

A006179 Witt vector *4!/4!.

Original entry on oeis.org

1, 52, 5133, 655554, 97772875, 16019720210, 2812609211657, 518332479161091, 99318252448110232, 19600890528520952329, 3966181169996511862429, 818653886943854653597621, 171938262068874336023196923

Offset: 1

Simon Plouffe

nonn

If c is the Witt transform of b then b(n) = Sum_{d|n} A074650(n/d, c(d)).

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

H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-108.
H. Gaudier, Relèvement des coefficients binomiaux dans les vecteurs de Witt, Séminaire Lotharingien de Combinatoire. Institut de Recherche Math. Avancée, Université Louis Pasteur, Strasbourg, Actes 16 (1988), 358/S-18, pp. 93-107. (Annotated scanned copy)

Witt transform of A029809.

More terms and formula from Christian G. Bower, Aug 28 2002

cp's OEIS Frontend

A060221 Number of orbits of length n under the full 18-shift (whose periodic points are counted by A001027).

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A060222 Number of orbits of length n under the full 19-shift (whose periodic points are counted by A001029).

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A292350 Number of Lyndon words (aperiodic necklaces) with 6 beads of n colors.

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A352745 a(n) is the number of Lyndon factors of the Fibonacci string of length n.

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A352746 a(n) is the number of Lyndon factors of the Thue-Morse string of length n.

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A006174 Witt vector *3!.

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A006175 Witt vector *4!.

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A006176 Witt vector *5!.

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