A060165 -id:A060165 - cp's OEIS frontend

A060169 Number of orbits of length n under the automorphism of the 3-torus whose periodic points are counted by A001945.

1, 0, 0, 1, 0, 1, 1, 0, 2, 1, 2, 2, 2, 4, 4, 5, 8, 6, 12, 13, 16, 23, 26, 35, 46, 54, 76, 89, 120, 154, 192, 255, 322, 411, 544, 679, 898, 1145, 1476, 1925, 2466, 3201, 4156, 5338, 6978, 8985

Offset: 1

Thomas Ward, Mar 13 2001

The sequence A001945 records the number of points of period n under a map. The number of orbits of length n for this map gives the sequence above.

			u(17) = 8 since the map whose periodic points are counted by A001945 has 1 fixed point and 137 points of period 17, hence 8 orbits of length 7.

Manfred Einsiedler, Graham Everest and Thomas Ward, Primes in sequences associated to polynomials (after Lehmer), LMS J. Comput. Math. 3 (2000), 125-139.
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.

Cf. A001642, A060164, A060165, A060166, A060167, A060168, A060170, A060171, A060171, A060172, A060173.

a(n) = (1/n)* Sum_{ d divides n } mu(d)*A001945(n/d).

A268592 a(n) = (6/n^3) * Sum_{d|n} moebius(n/d)binomial(2d,d).

Original entry on oeis.org

12, 3, 4, 6, 12, 25, 60, 150, 400, 1107, 3180, 9386, 28404, 87711, 275764, 880470, 2849916, 9336508, 30918732, 103384758, 348725540, 1185630123, 4060210764, 13996354586, 48541672872, 169293988125, 593488622344, 2090567755278, 7396924802052, 26281018091013, 93738717046476, 335563502259798

Offset: 1

Max Alekseyev, Feb 07 2016

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1500
R. R. Aidagulov, M. A. Alekseyev. On p-adic approximation of sums of binomial coefficients. Journal of Mathematical Sciences 233:5 (2018), 626-634. doi:10.1007/s10958-018-3948-0 arXiv:1602.02632

Cf. A254593, A268618, A268619.

a[n_] := (6/n^3)* DivisorSum[n, MoebiusMu[n/#] Binomial[2 #, #] &]; Array[a, 50] (* G. C. Greubel, Dec 15 2017 *)

{ a(n) = sumdiv(n, d, moebius(n/d)*binomial(2*d, d))*6/n^3; }

a(n) = A007727(n)*6/n^3 = A045630(n)*12/n^3 = A060165(n)*6/n^2 = A022553(n)*12/n^2 = A268619(n)*6/n.

For n == 0, 1, or 3 (mod 4), a(n) = 2*A254593(n); for n == 2 (mod 4), a(n) = 2*A254593(n) - A254593(n/2)/2.

A060172 Number of orbits of length n under a map whose periodic points are counted by A027306.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 9, 19, 28, 62, 93, 205, 315, 703, 1091, 2440, 3855, 8616, 13797, 30801, 49929, 111311, 182361, 405751, 671088, 1490409, 2485504, 5509504, 9256395, 20480421, 34636833, 76499520, 130150493, 286960946, 490853403, 1080476338, 1857283155, 4081876927, 7048151355

Offset: 1

Thomas Ward, Mar 13 2001

The sequence A027306 records the number of points of period n under a map. The number of orbits of length n for this map gives the sequence above.

			u(7) = 9 since the map whose periodic points are counted by A027306 has 1 fixed point and 64 points of period 7, hence 9 orbits of length 7.

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.

Cf. A027306, A060164, A060165, A060166, A060167, A060168, A060169, A060170, A060171, A060173.

a027306(n) = (2^n + if(n%2, 0, binomial(n, n/2)))/2;
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a027306(n/d)); \\ Michel Marcus, Sep 11 2017

a(n) = (1/n)* Sum_{ d divides n } mu(d)*A027306(n/d).

More terms from Michel Marcus, Sep 11 2017

A060173 Number of orbits of length n under a map whose periodic points are counted by A056045.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 12, 10, 30, 1, 139, 1, 252, 231, 920, 1, 3780, 1, 10250, 5601, 32076, 1, 149390, 2126, 400036, 173692, 1475642, 1, 6196651, 1, 19113136, 5864915, 68635494, 201405, 289525026, 1, 930138540, 208267554, 3469290971, 1, 14075005210, 1, 47994721225, 7683440470

Offset: 1

Thomas Ward, Mar 13 2001

The sequence A056045 records the number of points of period n under a map. The number of orbits of length n for this map gives the sequence above.

			a(7) = 1 since the map whose periodic points are counted by A056045 has 1 fixed point and 8 points of period 7, hence 1 orbits of length 7.

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.

Cf. A056045, A060164, A060165, A060166, A060167, A060168, A060169, A060170, A060171, A060172.

a056045(n) = sumdiv(n, d, binomial(n, d));
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a056045(n/d)); \\ Michel Marcus, Sep 11 2017

a(n) = (1/n)* Sum_{ d divides n } mu(d)*A056045(n/d).

More terms from Michel Marcus, Sep 11 2017

A268619 a(n) = (1/n^2) * Sum_{d|n} moebius(n/d)binomial(2d,d).

Original entry on oeis.org

2, 1, 2, 4, 10, 25, 70, 200, 600, 1845, 5830, 18772, 61542, 204659, 689410, 2347920, 8074762, 28009524, 97909318, 344615860, 1220539390, 4347310451, 15564141262, 55985418344, 202256970300, 733607281875, 2670698800548, 9755982857964, 35751803209918, 131405090455065, 484316704740126, 1789672012052256

Offset: 1

Max Alekseyev, Feb 09 2016

nonn

6*a(n) is divisible by n (cf. A268592).

Cf. A000984, A007727, A008683, A022553, A045630, A060165, A268592, A268617.

a[n_] := DivisorSum[n, MoebiusMu[n/#] * Binomial[2*#, #] &] / n^2; Array[a, 35] (* Amiram Eldar, Aug 24 2023 *)

{ a(n) = sumdiv(n, d, moebius(n/d)*binomial(2*d, d))/n^2; }

a(n) = (1/n^2)* Sum_{d|n} A008683(n/d)*A000984(d).

a(n) = A007727(n)/n^2 = A045630(n)*2/n^2 = A060165(n)/n = A022553(n)*2/n.

A349001 The number of Lyndon words of size n from an alphabet of 5 letters and 1st and 2nd letter of the alphabet with equal frequency in the words.

Original entry on oeis.org

1, 3, 4, 14, 46, 174, 656, 2640, 10790, 45340, 193600, 839820, 3686424, 16353924, 73187456, 330052646, 1498335650, 6841899606, 31404443032, 144814450188, 670552118244, 3116578216310, 14534401932712, 67992210407514, 318969964124256, 1500268062754830

Offset: 0

R. J. Mathar, Nov 05 2021

nonn

Counts a subset of the Lyndon words in A001692. Here there is no requirement of how often the 3rd to 5th letter of the alphabet are in the admitted word, only on the frequency of the 1st and 2nd letter of the alphabet.

Let T(n,k,M) be the number of words of length n drawn from an alphabet of size M where the first k letters of the alphabet appear with the same frequency f in each word. Then T(n,k,M) = Sum_{f=0..floor(n/k)} (M-k)^(n-f*k) * Product_{i=0..k-1} binomial(n-i*f,f) and T(n,2,5) = A026375(n), T(n,3,6) = A294035(n), T(n,2,6) = A081671(n). Removing the words with cycles by the inclusion-exclusion principle by a Mobius Transform gives words of length n of that type without cycles and division through n the Lyndon words of that type. - R. J. Mathar, Nov 07 2021

			Examples for the alphabet {0,1,2,3,4}:
a(0)=1 counts (), the empty word.
a(3)=14 counts (021) (031) (041) (012) (013) (223) (233) (243) (014) (224) (234) (334) (244) (344), words of length 3 where the letters 0 and the 1 occur both either not or once.
a(4)=46 counts (0011) (0221) (0321) (0421) (0231) (0331) (0431) (0241) (0341) (0441) (0212) (0312) (0412) (0122) (0132) (0142) (0213) (0313) (0413) (0123) (2223) (0133) (2233) (2333) (2433) (0143) (2243) (2343) (2443) (0214) (0314) (0414) (0124) (2224) (2324) (0134) (2234) (2334) (3334) (2434) (0144) (2244) (2344) (3344) (2444) (3444).

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for sequences related to Lyndon words

Cf. A022553 (alphabet of 2 letters), A290277 (of 3 letters), A060165 (of 4 letters), A026375.

a(n) = if(n>0, sumdiv(n, d, moebius(n/d)*sum(k=0, d, binomial(d,k)*binomial(2*k,k)))/n, n==0) \\ Andrew Howroyd, Jan 14 2023

n*a(n) = Sum_{d|n} mu(d)*A026375(n/d) where mu = A008683.

Terms a(16) and beyond from Andrew Howroyd, Jan 14 2023

A349002 The number of Lyndon words of size n from an alphabet of 4 letters and 1st, 2nd and 3rd letter of the alphabet with equal frequency in the words.

Original entry on oeis.org

1, 1, 0, 2, 6, 12, 34, 120, 354, 1082, 3636, 12270, 40708, 139062, 484866, 1692268, 5944470, 21134808, 75625330, 271720146, 982116648, 3569558058, 13025614962, 47714385708, 175470892468, 647508620070, 2396613522804, 8896422981608, 33114570409896, 123566641829256

Offset: 0

R. J. Mathar, Nov 05 2021

nonn

Counts a subset of the Lyndon words in A027377. Here there is no requirement of how often the 4th letter of the alphabet occurs in the admitted word, only on the frequency of the 1st to 3rd letter of the alphabet.

			Examples for the alphabet {0,1,2,3}:
a(0)=1 counts (), the empty word.
a(3)=2 counts (021) (012).
a(4)=6 counts (0321) (0231) (0312) (0132) (0213) (0123).
a(5)=12 counts (03321) (03231) (02331) (03312) (03132) (01332) (03213) (02313) (03123) (01323) (02133) (01233).
a(6)=34 counts (020211) (002211) (012021) (002121) (010221) (001221) (033321) (033231) (032331) (023331) (012102) (011202) (002112) (010212) (001212) (033312) (011022) (010122) (001122) (033132) (031332) (013332) (033213) (032313) (023313) (033123) (031323) (013323) (032133) (023133) (031233) (013233) (021333) (012333).

Index entries for sequences related to Lyndon words

Cf. A027377, A060165, A344560.

a(n) = if(n>0, sumdiv(n, d, moebius(n/d)*sum(k=0, d\3, d!/(k!^3*(d-3*k)!)))/n, n==0) \\ Andrew Howroyd, Jan 14 2023

n*a(n) = Sum_{d|n} mu(d)*A344560(n/d) where mu = A008683.

Terms corrected and extended by Andrew Howroyd, Jan 14 2023

cp's OEIS Frontend

A060169 Number of orbits of length n under the automorphism of the 3-torus whose periodic points are counted by A001945.

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A268592 a(n) = (6/n^3) * Sum_{d|n} moebius(n/d)*binomial(2*d,d).

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A060172 Number of orbits of length n under a map whose periodic points are counted by A027306.

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A060173 Number of orbits of length n under a map whose periodic points are counted by A056045.

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A268619 a(n) = (1/n^2) * Sum_{d|n} moebius(n/d)*binomial(2*d,d).

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A349001 The number of Lyndon words of size n from an alphabet of 5 letters and 1st and 2nd letter of the alphabet with equal frequency in the words.

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A349002 The number of Lyndon words of size n from an alphabet of 4 letters and 1st, 2nd and 3rd letter of the alphabet with equal frequency in the words.

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Examples

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Programs

PARI

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Extensions

A268592 a(n) = (6/n^3) * Sum_{d|n} moebius(n/d)binomial(2d,d).

A268619 a(n) = (1/n^2) * Sum_{d|n} moebius(n/d)binomial(2d,d).