A280218 -id:A280218 - cp's OEIS frontend

A000358 Number of binary necklaces of length n with no subsequence 00, excluding the necklace "0".

1, 2, 2, 3, 3, 5, 5, 8, 10, 15, 19, 31, 41, 64, 94, 143, 211, 329, 493, 766, 1170, 1811, 2787, 4341, 6713, 10462, 16274, 25415, 39651, 62075, 97109, 152288, 238838, 375167, 589527, 927555, 1459961, 2300348, 3626242, 5721045, 9030451, 14264309, 22542397, 35646312, 56393862, 89264835, 141358275

Offset: 1

Frank Ruskey

a(n) is also the number of inequivalent compositions of n into parts 1 and 2 where two compositions are considered to be equivalent if one is a cyclic rotation of the other. a(6)=5 because we have: 2+2+2, 2+2+1+1, 2+1+2+1, 2+1+1+1+1, 1+1+1+1+1+1. - Geoffrey Critzer, Feb 01 2014

Moebius transform is A006206. - Michael Somos, Jun 02 2019

			G.f. = x + 2*x^2 + 2*x^3 + 3*x^4 + 3*x^5 + 5*x^6 + 5*x^7 + 8*x^8 + 10*x^9 + ... - _Michael Somos_, Jun 02 2019
Binary necklaces are: 1; 01, 11; 011, 111; 0101, 0111, 1111; 01010, 01011, 01111. - _Michael Somos_, Jun 02 2019

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 499.
T. Helleseth and A. Kholosha, Bent functions and their connections to combinatorics, in Surveys in Combinatorics 2013, edited by Simon R. Blackburn, Stefanie Gerke, Mark Wildon, Camb. Univ. Press, 2013.

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Ricardo Gómez Aíza, Symbolic dynamical scales: modes, orbitals, and transversals, arXiv:2009.02669 [math.DS], 2020.
A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar, and Marko Petkovsek, Orbits of Fibonacci and Lucas cubes, dihedral transformations, and asymmetric strings, 2014.
A. R. Ashrafi, J. Azarija, K. Fathalikhani, S. Klavzar, and Marko Petkovsek, Vertex and Edge Orbits of Fibonacci and Lucas Cubes, Ann. Comb. 20 (2016), 209-229.
M. Assis, J. L. Jacobsen, I. Jensen, J.-M. Maillard, and B. M. McCoy, Integrability vs non-integrability: Hard hexagons and hard squares compared, arXiv preprint 1406.5566 [math-ph], 2014.
M. Assis, J. L. Jacobsen, I. Jensen, J.-M. Maillard, and B. M. McCoy, Integrability versus non-integrability: Hard hexagons and hard squares compared, J. Phys. A: Math. Theor. 47 (2014) 445001 (53pp.).
Daryl DeFord, Enumerating distinct chessboard tilings, Fibonacci Quart. 52 (2014), 102-116; see formula (5.3) in Theorem 5.2, p. 111.
P. Flajolet and M. Soria, The Cycle Construction, SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.
P. Flajolet and M. Soria, The Cycle Construction, SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.
Benjamin Hackl and Helmut Prodinger, The Necklace Process: A Generating Function Approach, arXiv:1801.09934 [math.PR], 2018. [The paper mentions this sequence, but the authors mean sequence A032190(n) = a(n) - 1.]
Benjamin Hackl and Helmut Prodinger, The Necklace Process: A Generating Function Approach, Statistics and Probability Letters 142 (2018), 57-61. [The paper mentions this sequence, but the authors mean sequence A032190(n) = a(n) - 1.]
P. Hadjicostas, Cyclic compositions of a positive integer with parts avoiding an arithmetic sequence, J. Integer Sequences 19 (2016), #16.8.2.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 119
Tom Roby, Dynamical algebraic combinatorics and homomesy: An action-packed introduction, AlCoVE: an Algebraic Combinatorics Virtual Expedition (2020).
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
C. J. Turner, A. A. Michailidis, D. A. Abanin, M. Serbyn, and Z. Papić, Quantum scarred eigenstates in a Rydberg atom chain: entanglement, breakdown of thermalization, and stability to perturbations, arXiv:1806.10933 [cond-mat.quant-gas], 2018.
L. Zhang and P. Hadjicostas, On sequences of independent Bernoulli trials avoiding the pattern '11..1', Math. Scientist, 40 (2015), 89-96.
Index entries for sequences related to necklaces

Cf. A006206, A032190, A093305, A280218, A280218.

Column k=0 of A320341.

A000358 := proc(n) local sum; sum := 0; for d in divisors(n) do sum := sum + phi(n/d)*(fibonacci(d+1)+fibonacci(d-1)) od; RETURN(sum/n); end;
with(combstruct); spec := {A=Union(zero,Cycle(one),Cycle(Prod(zero,Sequence(one,card>0)))),one=Atom,zero=Atom}; seq(count([A,spec,unlabeled],size=i),i=1..30);

nn=48;Drop[Map[Total,Transpose[Map[PadRight[#,nn]&,Table[ CoefficientList[ Series[CycleIndex[CyclicGroup[n],s]/.Table[s[i]->x^i+x^(2i),{i,1,n}],{x,0,nn}],x],{n,0,nn}]]]],1] (* Geoffrey Critzer, Feb 01 2014 *)
max = 50; B[x_] := x*(1+x); A = Sum[EulerPhi[k]/k*Log[1/(1-B[x^k])], {k, 1, max}]/x + O[x]^max; CoefficientList[A, x] (* Jean-François Alcover, Feb 08 2016, after Joerg Arndt *)
Table[1/n * Sum[EulerPhi[n/d] Total@ Map[Fibonacci, d + # & /@ {-1, 1}], {d, Divisors@ n}], {n, 47}] (* Michael De Vlieger, Dec 28 2016 *)
a[ n_] := If[ n < 1, 0, DivisorSum[n, EulerPhi[n/#] LucasL[#] &]/n]; (* Michael Somos, Jun 02 2019 *)

N=66;  x='x+O('x^N);
B(x)=x*(1+x);
A=sum(k=1, N, eulerphi(k)/k*log(1/(1-B(x^k))));
Vec(A)
/* Joerg Arndt, Aug 06 2012 */

{a(n) = if( n<1, 0, sumdiv(n, d, eulerphi(n/d) * (fibonacci(d+1) + fibonacci(d-1)))/n)}; /* Michael Somos, Jun 02 2019 */

from sympy import totient, lucas, divisors
def A000358(n): return (n&1^1)+sum(totient(n//k)*(lucas(k)-((k&1^1)<<1)) for k in divisors(n,generator=True))//n # Chai Wah Wu, Sep 23 2023

a(n) = (1/n) * Sum_{ d divides n } totient(n/d) [ Fib(d-1)+Fib(d+1) ].
G.f.: Sum_{k>=1} phi(k)/k * log( 1/(1-B(x^k)) ) where B(x)=x*(1+x). - Joerg Arndt, Aug 06 2012
a(n) ~ ((1+sqrt(5))/2)^n / n. - Vaclav Kotesovec, Sep 12 2014
a(n) = Sum_{0 <= i <= ceiling((n-1)/2)} [ (1/(n - i)) * Sum_{d|gcd(i, n-i)} phi(d) * binomial((n - i)/d, i/d) ]. (This is DeFord's formula for the number of distinct Lucas tilings of a 1 X n bracelet up to symmetry, even though in the paper he refers to sequence A032192(n) = a(n) - 1.) - Petros Hadjicostas, Jun 07 2019

A093305 Number of binary necklaces of length n with no subsequence 000.

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 11, 19, 29, 48, 75, 132, 213, 369, 627, 1083, 1857, 3244, 5619, 9844, 17205, 30229, 53115, 93701, 165313, 292464, 517831, 918578, 1630933, 2900109, 5161443, 9197251, 16402841, 29283026, 52319379, 93558968, 167427845, 299846737, 537358107, 963651447, 1729192433

Offset: 1

Philippe Deléham, Apr 24 2004

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 500.

Michael De Vlieger, Table of n, a(n) for n = 1..500
P. Flajolet and M. Soria, The Cycle Construction, SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.
P. Flajolet and M. Soria, The Cycle Construction, SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.
Petros Hadjicostas, Cyclic Compositions of a Positive Integer with Parts Avoiding an Arithmetic Sequence, Journal of Integer Sequences, 19 (2016), #16.8.2.
Silvana Ramaj, New Results on Cyclic Compositions and Multicompositions, Master's Thesis, Georgia Southern Univ., 2021. See p. 57.
L. Zhang and P. Hadjicostas, On sequences of independent Bernoulli trials avoiding the pattern '11..1', Math. Scientist, 40 (2015), 89-96.

Row 3 of A322057.

Cf. A000358, A001644, A280218, A280303.

Table[1/n * Sum[EulerPhi[n/d] (d Sum[Sum[Binomial[j, d - 3 k + 2 j] Binomial[k, j], {j, d - 3 k, k}]/k, {k, d}]), {d, Divisors@ n}], {n, 41}] (* Michael De Vlieger, Dec 28 2016, after Vladimir Joseph Stephan Orlovsky at A001644 *)

N=66;  x='x+O('x^N);
B(x)=x*(1+x+x^2);
A=sum(k=1, N, eulerphi(k)/k*log(1/(1-B(x^k))));
Vec(A)
/* Joerg Arndt, Aug 06 2012 */

a(n) = (1/n) * Sum_{d divides n} totient(n/d)*A001644(d).
G.f.: Sum_{k>=1} phi(k)/k * log( 1/(1-B(x^k)) ) where B(x) = x*(1+x+x^2). - Joerg Arndt, Aug 06 2012
a(n) ~ d^n / n, where d = (19 + 3*sqrt(33))^(1/3)/3 + 4/(3*(19 + 3*sqrt(33))^(1/3)) + 1/3 = A058265 = 1.8392867552141611325518... - Vaclav Kotesovec, Jul 13 2019

A351361 Number of binary necklaces with n beads and at least four consecutive black beads.

Original entry on oeis.org

0, 0, 0, 1, 2, 3, 5, 9, 17, 33, 63, 124, 241, 475, 930, 1831, 3593, 7077, 13905, 27377, 53847, 106006, 208601, 410723, 808514, 1592123, 3135085, 6174695, 12161761, 23957945, 47198847, 92997768, 183253045, 361145269, 711793554, 1403055509, 2765913617

Offset: 1

Marko Riedel, Feb 08 2022

nonn

Marko R. Riedel et al., Necklaces made of black and white beads, Math StackExchange, Feb 2022.

Cf. A351356, A351357, A351358, A351359, A351360.

a(n) = A000031(n) - A280218(n).

A322057 Array read by upwards antidiagonals: T(i,n) is the number of binary necklaces of length n that avoid 00...0 (i 0's).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 3, 1, 1, 2, 3, 4, 3, 1, 1, 2, 3, 5, 5, 5, 1, 1, 2, 3, 5, 6, 9, 5, 1, 1, 2, 3, 5, 7, 11, 11, 8, 1, 1, 2, 3, 5, 7, 12, 15, 19, 10, 1, 1, 2, 3, 5, 7, 13, 17, 27, 29, 15, 1, 1, 2, 3, 5, 7, 13, 18, 31, 43, 48, 19, 1

Offset: 1

N. J. A. Sloane, Dec 25 2018

T(i,n) is the number of necklace compositions with sum n and parts at most i. For example, the T(3,5) = 5 compositions up to cyclic equivalence are 11111, 1112, 113, 122, 23. - Andrew Howroyd, Jan 08 2024

			The first few antidiagonals are:
  1;
  1,  1;
  1,  2,  1;
  1,  2,  2,  1;
  1,  2,  3,  3,  1;
  1,  2,  3,  4,  3,  1;
  1,  2,  3,  5,  5,  5,  1;
  1,  2,  3,  5,  6,  9,  5,  1;
  1,  2,  3,  5,  7, 11, 11,  8,  1;
  1,  2,  3,  5,  7, 12, 15, 19, 10,  1;
  ...
From _Petros Hadjicostas_, Jan 16 2019: (Start)
In the above triangle (first few antidiagonals, read upwards), the j-th row corresponds to T(j,1), T(j-1,2), T(j-2,3), ..., T(1,j).
This, however, is not the j-th row of the square array (see the scanned page above).
For example, the sixth row of the square array is as follows:
T(6,1) = 1, T(6,2) = 2, T(6,3) = 3, T(6,4) = 5, T(6, 5) = 7, T(6, 6) = 13, ...
To generate these numbers, we use T(6, n) = (1/n)*Sum_{d|n} phi(n/d)*L(6,d), where
L(6,1) = 1, L(6,2) = 3, L(6,3) = 7, L(6,4) = 15, L(6,5) = 31, L(6,6) = 63, ...
See the sixth row of A125127. See also the Sage program below by Freddy Barrera.
(End)

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 520.

Freddy Barrera, Rows n = 1..100 of triangle, flattened.
Miklos Bona, Handbook of Enumerative Combinatorics, CRC Press, 2015, annotated scan of page 520.
P. Flajolet and M. Soria, The Cycle Construction, SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.
L. Zhang and P. Hadjicostas, On sequences of independent Bernoulli trials avoiding the pattern '11..1', Math. Scientist, 40 (2015), 89-96.

Rows 2, 3, 4 and 5 are A000358, A093305, A280218, A280303.
The rows converge to A008965.
Cf. A119458.

T(i,n) = {my(p=1/(1 - x*(1 - x^i)/(1 - x))); polcoef(sum(d=1, n, eulerphi(d)*log(subst(p + O(x*x^(n\d)), x, x^d))/d), n)} \\ Andrew Howroyd, Jan 08 2024

# uses the L method from A125127
def T(i, n):
    return sum(euler_phi(n//d)*L(i, d) for d in n.divisors()) // n
[T(i, n) for d in (1..12) for i, n in zip((d..1, step=-1), (1..d))] # Freddy Barrera, Jan 15 2019

T(i,n) = (1/n) * Sum_{d divides n} totient(n/d)*L(i,d), where L(i,d) = A125127(i,d). See Zhang and Hadjicostas link. - Freddy Barrera, Jan 15 2019

G.f. for row i: Sum_{k>=1} (phi(k)/k) * log(1/(1-B(i, x^k))) where B(i, x) = x*(1 + x + x^2 + ... + x^(i-1)). (This is a generalization of Joerg Arndt's formulae for the g.f.'s of rows 2 and 3.) - Petros Hadjicostas, Jan 24 2019

A280303 Number of binary necklaces of length n with no subsequence 00000.

Original entry on oeis.org

1, 2, 3, 5, 7, 12, 17, 31, 51, 91, 155, 287, 505, 930, 1695, 3129, 5759, 10724, 19913, 37239, 69643, 130745, 245715, 463099, 873705, 1651838, 3126707, 5927817, 11251031, 21382558, 40679233, 77475673, 147694719, 281822847, 538213671, 1028714071, 1967728553

Offset: 1

Petros Hadjicostas and Lingyun Zhang, Dec 31 2016

nonn

a(n) is the number of cyclic sequences of length n consisting of zeros and ones that do not contain five consecutive zeros provided we consider as equivalent those sequences that are cyclic shifts of each other.

			a(5)=7 because we have seven binary cyclic sequences (necklaces) of length 5 that avoid five consecutive zeros: 00001, 00011, 00101, 00111, 01101, 01111, 11111.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
P. Flajolet and M. Soria, The Cycle Construction, SIAM J. Discr. Math., vol. 4 (1), 1991, pp. 58-60.
Petros Hadjicostas, Proof of the formula for the generating function from the formula for a(n)
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
L. Zhang and P. Hadjicostas, On sequences of independent Bernoulli trials avoiding the pattern '11..1', Math. Scientist, 40 (2015), 89-96.

Row 5 of A322057.

Cf. A000358, A093305, A280218, A074048.

a(n) = (1/n) * Sum_{d divides n} totient(n/d) * A074048(d).

G.f.: Sum_{k>=1} (phi(k)/k) * log(1/(1-B(x^k))) where B(x) = x*(1+x+x^2+x^3+x^4).

a(34) onwards from Andrew Howroyd, Jan 25 2024

A063686 Triangular array: T(n,k) is the number of binary necklaces (no turning over) of length n whose longest run of 1's has length k. Table begins at n=0, k=0.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 4, 4, 2, 1, 1, 1, 1, 4, 6, 4, 2, 1, 1, 1, 1, 7, 11, 8, 4, 2, 1, 1, 1, 1, 9, 19, 14, 8, 4, 2, 1, 1, 1, 1, 14, 33, 27, 16, 8, 4, 2, 1, 1, 1, 1, 18, 56, 50, 30, 16, 8, 4, 2, 1, 1, 1, 1, 30, 101, 96, 59, 32, 16, 8, 4, 2, 1, 1, 1

Offset: 0

Christopher Lenard (c.lenard(AT)bendigo.latrobe.edu.au), Aug 22 2001

Column k=1 appears to be A032190(n), n=2,3,...

			Triangle begins:
  1;
  1, 1;
  1, 1, 1;
  1, 1, 1, 1;
  1, 2, 1, 1, 1;
  1, 2, 2, 1, 1, 1;
  1, 4, 4, 2, 1, 1, 1;
  1, 4, 6, 4, 2, 1, 1, 1;
  1, 7, 11, 8, 4, 2, 1, 1, 1;
  1, 9, 19, 14, 8, 4, 2, 1, 1, 1;
  1, 14, 33, 27, 16, 8, 4, 2, 1, 1, 1;
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1274 (Rows n=0..49)

Cf. A032190, A048004, A048887.

Cf. A000358, A093305, A280218 (necklaces avoiding 00, 000, 0000).

\\ here R(n) is A048887 transposed
R(n)={Mat(vector(n, k, Col((1-x)/(1-2*x+x^(k+1)) - 1 + O(x*x^n))))}
S(M)={matrix(#M-1, #M-1, n, k, if(kAndrew Howroyd, Oct 15 2017

T(0,0)=1 from Andrew Howroyd, Oct 15 2017

cp's OEIS Frontend

A000358 Number of binary necklaces of length n with no subsequence 00, excluding the necklace "0".

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A093305 Number of binary necklaces of length n with no subsequence 000.

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A351361 Number of binary necklaces with n beads and at least four consecutive black beads.

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A322057 Array read by upwards antidiagonals: T(i,n) is the number of binary necklaces of length n that avoid 00...0 (i 0's).

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A280303 Number of binary necklaces of length n with no subsequence 00000.

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A063686 Triangular array: T(n,k) is the number of binary necklaces (no turning over) of length n whose longest run of 1's has length k. Table begins at n=0, k=0.

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