A000011 -id:A000011 - cp's OEIS frontend

A019536 Number of length n necklaces with integer entries that cover an initial interval of positive integers.

1, 2, 5, 20, 109, 784, 6757, 68240, 787477, 10224812, 147512053, 2340964372, 40527565261, 760095929840, 15352212731933, 332228417657960, 7668868648772701, 188085259070219000, 4884294069438337429

Offset: 1

Manfred Goebel (goebel(AT)informatik.uni-tuebingen.de)

Original name: a(n) = number of necklaces of n beads with up to n unlabeled colors.

The Moebius transform of this sequence is A060223.

			a(3) = 5 since there are the following length 3 words up to rotation:
     111,  112, 122, 123, 132.
a(4) = 20 since there are the following length 4 words up to rotation:
     1111,
     1112, 1122, 1212, 1222,
     1123, 1132, 1213, 1223, 1232, 1233, 1322, 1323, 1332,
     1234, 1243, 1324, 1342, 1423, 1432.

G. C. Greubel, Table of n, a(n) for n = 1..420
M. Goebel, On the number of special permutation-invariant orbits and terms, in: Applicable Algebra in Engin., Comm. and Comp. (AAECC 8), Volume 8, Number 6, 1997, pp. 505-509 (Lect. Notes Comp. Sci.); see p. 509 (stated as an open problem).
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]
Eric Weisstein's world of Mathematics, Necklaces.

Cf. A000670, A008277, A019537, A060223.

Row sums of A087854. - Philippe Deléham

Needs["DiscreteMath`Combinatorica`"];
mult[li:{__Integer}] := Multinomial @@ Length /@ Split[Sort[li]];
neck[li:{__Integer}] := Module[{n, d}, n=Plus @@ li; d=n-First[li];Fold[ #1+(EulerPhi[ #2]*(n/#2)!)/Times @@ ((li/#2)!)&, 0, Divisors[GCD @@ li]]/n];
Table[(mult /@ Partitions[n]).(neck /@ Partitions[n]), {n, 24}]
(* second program: *)
a[n_] := Sum[DivisorSum[n, EulerPhi[#]*StirlingS2[n/#, k] k! &]/n, {k, 1, n}];
Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Mar 31 2016, after Philippe Deléham *)

a(n) = sum(k=1, n, sumdiv(n, d, eulerphi(d)*stirling(n/d, k, 2)*k!)/n); \\ Michel Marcus, Mar 31 2016

See Mathematica code.
a(n) ~ (n-1)! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Jul 21 2019
From Petros Hadjicostas, Aug 19 2019: (Start)
The first formula is due to Philippe Deléham from the Crossrefs (see also the programs below). The second one follows easily from the first one. The third one follows from the second one using the associative property of Dirichlet convolutions.
a(n) = Sum_{k = 1..n} (k!/n) * Sum_{d|n} phi(d) * S2(n/d, k), where S2(n, k) = Stirling numbers of 2nd kind (A008277).
a(n) = (1/n) * Sum_{d|n} phi(d) * A000670(n/d).
a(n) = Sum_{d|n} A060223(d).
(End)
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = (1/n)*Sum_{k=1..n} A000670(gcd(n,k)).
a(n) = (1/n)*Sum_{k=1..n} A000670(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

Edited by Wouter Meeussen, Aug 06 2002
Corrected by T. D. Noe, Oct 31 2006
Edited by Andrew Howroyd, Aug 19 2019

A000013 Definition (1): Number of n-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed.

Original entry on oeis.org

1, 1, 2, 2, 4, 4, 8, 10, 20, 30, 56, 94, 180, 316, 596, 1096, 2068, 3856, 7316, 13798, 26272, 49940, 95420, 182362, 349716, 671092, 1290872, 2485534, 4794088, 9256396, 17896832, 34636834, 67110932, 130150588, 252648992, 490853416, 954444608, 1857283156, 3616828364

Offset: 0

N. J. A. Sloane

Definition (2): Equivalently, number of different output sequences from an n-stage pure cycling shift register when 2 sequences are considered the same if one is the complement of the other.
Definition (3): Also number of different output sequences from an n-stage pure cycling shift register constrained so contents have even weight.
Definition (4): Also number of output sequences from (n-1)-stage shift register which feeds back the mod 2 sum of the contents of the register.
The equivalence of definitions (1) and (2) follows at once from the definitions.
If u is an output sequence of type (2) then its derivative is of type (3) - so (2) and (3) count the same things.
If we have a shift register of type (4), append a new cell which contains the mod 2 sum of the contents to get a shift register of type (3). So (3) and (4) count the same things.
If n is even, a(n) = A000116(n/2). If 2^(n+1)-1 is prime, then a(n) = A128976(n+1), the number of cycles in the digraph of the Lucas-Lehmer operator LL(x) = x^2 - 2 acting on Z/(2^(n+1)-1). - M. F. Hasler, May 19 2007
Also number of 2n-bead balanced binary necklaces that are equivalent to their complements. - Andrew Howroyd, Sep 29 2017

			G.f. = 1 + x + 2*x^2 + 2*x^3 + 4*x^4 + 4*x^5 + 8*x^6 + 10*x^7 + 20*x^8 + ...

S. W. Golomb, Shift-Register Sequences, Holden-Day, San Francisco, 1967, p. 172.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..3334 (first 201 terms from T. D. Noe)
Nicolás Álvarez, Victória Becher, Martín Mereb, Ivo Pajor, and Carlos Miguel Soto, On extremal factors of de Bruijn-like graphs, Univ. Buenos Aires (Argentina 2023). See also arXiv:2308.16257 [math.CO], 2023. See references.
Joerg Arndt, Matters Computational (The Fxtbook), p. 151, p. 408.
Raghav Bhutani and Frederick Saia, Replacement dynamics of binary quadratic forms, arXiv:2508.05816 [math.NT], 2025. See p. 6.
Henry Bottomley, Initial terms of A000011 and A000013
Zachary E. Chin and Isaac L. Chuang, The quantum trajectory sensing problem and its solution, arXiv:2410.00893 [quant-ph], 2024. See p. 19.
N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Darij Grinberg and Peter Mao, Necklaces over a group with identity product, arXiv:2405.08937 [math.CO], 2024. See pp. 15, 22.
Karyn McLellan, Periodic coefficients and random Fibonacci sequences, Electronic Journal of Combinatorics, 20(4), 2013, #P32.
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
N. J. A. Sloane, On single-deletion-correcting codes
N. J. A. Sloane, On single-deletion-correcting codes, arXiv:math/0207197 [math.CO], 2002; in Codes and Designs (Columbus, OH, 2000), 273-291, Ohio State Univ. Math. Res. Inst. Publ., 10, de Gruyter, Berlin, 2002.
N. J. A. Sloane, Maple code for this and related sequences
Index entries for sequences related to necklaces

Cf. A000031, A000016, A000116.
Cf. A128976.
Cf. A000010, A027750.
Cf. A385665, A000048.

a000013 0 = 1
a000013 n = sum (zipWith (*)
   (map (a000010 . (* 2)) ds) (map (2 ^) $ reverse ds)) `div` (2 * n)
   where ds = a027750_row n
-- Reinhard Zumkeller, Jul 08 2013

with(numtheory): A000013 := proc(n) local s,d; if n = 0 then RETURN(1) else s := 0; for d in divisors(n) do s := s+(phi(2*d)*2^(n/d))/(2*n); od; RETURN(s); fi; end;

a[n_] := Fold[ #1 + EulerPhi[2#2]2^(n/#2)/(2n) &, 0, Divisors[n]]
a[ n_] := If[ n < 1, Boole[n == 0], DivisorSum[ n, EulerPhi[2 #] 2^(n/#) &] / (2 n)]; (* Michael Somos, Dec 19 2014 *)
mx=40;CoefficientList[Series[1-Sum[EulerPhi[2i] Log[1-2*x^i]/(2i),{i,1,mx}],{x,0,mx}],x] (* Herbert Kociemba, Nov 01 2016 *)

{a(n) = if( n<1, n==0, sumdiv(n, k, eulerphi(2*k) * 2^(n/k)) / (2*n))}; /* Michael Somos, Oct 20 1999 */

from sympy import divisors, totient
def a(n): return 1 if n<1 else sum([totient(2*d)*2**(n//d) for d in divisors(n)])//(2*n) # Indranil Ghosh, Apr 28 2017

a(n) = Sum_{ d divides n } (phi(2*d)*2^(n/d))/(2*n) for n>0. - Michael Somos, Oct 20 1999
G.f.: 1 - Sum_{i>=1} phi(2*i)*log(1-2*x^i)/(2*i). - Herbert Kociemba, Nov 01 2016
From Richard L. Ollerton, May 11 2021: (Start)
For n >= 1:
a(n) = (1/(2*n))*Sum_{k=1..n} phi(2*gcd(n,k))*2^(n/gcd(n,k))/phi(n/gcd(n,k)), where phi = A000010.
a(n) = (1/(2*n))*Sum_{k=1..n} phi(2*n/gcd(n,k))*2^gcd(n,k)/phi(n/gcd(n,k)). (End)
a(n) ~ 2^(n-1)/n. - Cedric Lorand, Apr 24 2022
a(n) = Sum_{k=1..n} A385665(n,k) = Sum_{d|n} A000048(d). - Tilman Piesk, Jul 31 2025

A000029 Number of necklaces with n beads of 2 colors, allowing turning over (these are also called bracelets).

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 13, 18, 30, 46, 78, 126, 224, 380, 687, 1224, 2250, 4112, 7685, 14310, 27012, 50964, 96909, 184410, 352698, 675188, 1296858, 2493726, 4806078, 9272780, 17920860, 34669602, 67159050, 130216124, 252745368, 490984488, 954637558, 1857545300

Offset: 0

N. J. A. Sloane

"Necklaces with turning over allowed" are usually called bracelets. - Joerg Arndt, Jun 10 2016

			For n=2, the three bracelets are AA, AB, and BB. For n=3, the four bracelets are AAA, AAB, ABB, and BBB. - _Robert A. Russell_, Sep 24 2018

J. L. Fisher, Application-Oriented Algebra (1977), ISBN 0-7002-2504-8, circa p. 215.
Martin Gardner, "New Mathematical Diversions from Scientific American" (Simon and Schuster, New York, 1966), pages 245-246.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. Zagaglia Salvi, Ordered partitions and colourings of cycles and necklaces, Bull. Inst. Combin. Appl., 27 (1999), 37-40.

N. J. A. Sloane, Table of n, a(n) for n = 0..300
Hoda Abbasizanjani and Oliver Kullmann, Classification of minimally unsatisfiable 2-CNFs, arXiv:2003.03639 [cs.DM], 2020.
Joerg Arndt, Matters Computational (The Fxtbook), p. 151
Henry Bottomley, Illustration of initial terms
Emanuele Brugnoli, Enumerating the Walecki-Type Hamiltonian Cycle Systems, Journal of Combinatorial Designs, Volume 25, Issue 11, November 2017, pp. 481-493.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
Vladimir Dotsenko and Irvin Roy Hentzel, On the conjecture of Kashuba and Mathieu about free Jordan algebras, arXiv:2507.00437 [math.RA], 2025. See p. 14.
S. N. Ethier and J. Lee, Parrondo games with spatial dependence, arXiv preprint arXiv:1202.2609 [math.PR], 2012. - From _N. J. A. Sloane_, Jun 10 2012
S. N. Ethier and J. Lee, Parrondo games with spatial dependence II, Fluctuation and Noise Letters 11 (4) (2012), 1250030.
S. N. Ethier, Counting toroidal binary arrays, arXiv preprint arXiv:1301.2352 [math.CO], 2013.
S. N. Ethier, Counting toroidal binary arrays, Journal of Integer Sequences 16 (2013), Article 13.4.7.
N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Jurij Kovič, Regular polygonal systems, Ars Mathematica Contemporanea (2019) Vol. 16, No. 2, 157-171.
Jia Liu, L. Lalouat, E. Drouard, and R. Orobtchouk, Binary coded patterns for photon control using necklace problem concept, Optics Express Vol. 24, Issue 2, pp. 1133-1142 (2016).
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Zhe Sun, T. Suenaga, P. Sarkar, S. Sato, M. Kotani, and H. Isobe, Stereoisomerism, crystal structures, and dynamics of belt-shaped cyclonaphthylenes, Proc. Nat. Acad. Sci. USA, vol. 113 no. 29, pp. 8109-8114, doi: 10.1073/pnas.1606530113.
James Tilley, Stan Wagon, and Eric Weisstein, A Catalog of Facially Complete Graphs, arXiv:2409.11249 [math.CO], 2024. See p. 11.
A. M. Uludag, A. Zeytin and M. Durmus, Binary Quadratic Forms as Dessins, 2012. - From _N. J. A. Sloane_, Dec 31 2012
Eric Weisstein's World of Mathematics, Necklace
Eric Weisstein's World of Mathematics, e
Index entries for "core" sequences
Index entries for sequences related to bracelets
Index entries for sequences related to necklaces

Row sums of triangle in A052307, second column of A081720, column 2 of A051137.
Cf. A000011, A000013, A000031 (turning over not allowed), A001371 (primitive necklaces), A059076, A164090.
Cf. A000031, A051137.

with(numtheory): A000029 := proc(n) local d,s; if n = 0 then return 1 else if n mod 2 = 1 then s := 2^((n-1)/2) else s := 2^(n/2-2)+2^(n/2-1) fi; for d in divisors(n) do s := s+phi(d)*2^(n/d)/(2*n) od; return s; fi end:

a[0] := 1; a[n_] := Fold[#1 + EulerPhi[#2]2^(n/#2)/(2n) &, If[OddQ[n], 2^((n - 1)/2), 2^(n/2 - 1) + 2^(n/2 - 2)], Divisors[n]]
mx=40;CoefficientList[Series[(1-Sum[ EulerPhi[n]*Log[1-2*x^n]/n,{n,mx}]+(1+x)^2/(1-2*x^2))/2,{x,0,mx}],x] (* Herbert Kociemba, Nov 02 2016 *)
a[0] = 1; a[n_] := (1/4)*(Mod[n, 2] + 3)*2^Quotient[n, 2] + DivisorSum[n, EulerPhi[#]*2^(n/#)&]/(2*n); Array[a, 36, 0] (* Jean-François Alcover, Nov 05 2017 *)

a(n)=if(n<1,!n,(n%2+3)/4*2^(n\2)+sumdiv(n,d,eulerphi(n/d)*2^d)/2/n)

from sympy import divisors, totient
def a(n):
    return 1 if n<1 else ((2**(n//2+1) if n%2 else 3*2**(n//2-1)) + sum(totient(n//d)*2**d for d in divisors(n))//n)//2
print([a(n) for n in range(51)]) # Indranil Ghosh, Apr 23 2017

a(n) = Sum_{d divides n} phi(d)*2^(n/d)/(2*n) + either 2^((n - 1)/2) if n odd or 2^(n/2 - 1) + 2^(n/2 - 2) if n even.
G.f.: (1 - Sum_{n>=1} phi(n)*log(1 - 2*x^n)/n + (1 + x)^2/(1 - 2*x^2))/2. - Herbert Kociemba, Nov 02 2016
Equals (A000031 + A164090) / 2 = A000031 - A059076 = A059076 + A164090. - Robert A. Russell, Sep 24 2018
From Richard L. Ollerton, May 04 2021: (Start)
a(0) = 1; a(n) = Sum_{k=1..n} 2^gcd(n,k)/(2*n) + either 2^((n - 1)/2) if n odd or 2^(n/2 - 1) + 2^(n/2 - 2) if n even.
a(0) = 1; a(n) = A000031(n)/2 + (2^floor((n+1)/2) + 2^ceiling((n+1)/2))/4. See A051137. (End)

More terms from Christian G. Bower

A047996 Triangle read by rows: T(n,k) is the (n,k)-th circular binomial coefficient, where 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 4, 3, 1, 1, 1, 1, 3, 5, 5, 3, 1, 1, 1, 1, 4, 7, 10, 7, 4, 1, 1, 1, 1, 4, 10, 14, 14, 10, 4, 1, 1, 1, 1, 5, 12, 22, 26, 22, 12, 5, 1, 1, 1, 1, 5, 15, 30, 42, 42, 30, 15, 5, 1, 1, 1, 1, 6, 19, 43, 66, 80, 66, 43, 19, 6, 1, 1, 1, 1, 6, 22

Offset: 0

N. J. A. Sloane

Equivalently, T(n,k) = number of necklaces with k black beads and n-k white beads (binary necklaces of weight k).
The same sequence arises if we take the table U(n,k) = number of necklaces with n black beads and k white beads and read it by antidiagonals (cf. A241926). - Franklin T. Adams-Watters, May 02 2014
U(n,k) is also equal to the number of ways to express 0 as a sum of k elements in Z/nZ. - Jens Voß, Franklin T. Adams-Watters, N. J. A. Sloane, Apr 30 2014 - May 05 2014. See link ("A Note on Modular Partitions and Necklaces") for proof.
The generating function for column k is given by the substitution x_j -> x^j/(1-x^j) in the cycle index of the Symmetric Group of order k. - R. J. Mathar, Nov 15 2018
From Petros Hadjicostas, Jul 12 2019: (Start)
Regarding the comments above by Voss, Adams-Watters, and Sloane, note that Fredman (1975) proved that the number S(n, k, v) of vectors (a_0, ..., a_{n-1}) of nonnegative integer components that satisfy a_0 + ... + a_{n-1} = k and Sum_{i=0..n-1} i*a_i = v (mod n) is given by S(n, k, v) = (1/(n + k)) * Sum_{d | gcd(n, k)} A054535(d, v) * binomial((n + k)/d, k/d) = S(k, n, v).
This result was also proved by Elashvili et al. (1999), who also proved that S(n, k, v) = Sum_{d | gcd(n, k, v)} S(n/d, k/d, 1). Here, S(n, k, 0) = A241926(n, k) = U(n, k) = T(n + k, k) (where T(n, k) is the current array). Also, S(n, k, 1) = A245558(n, k). See also Panyushev (2011) for more general results and for generating functions.
Finally, note that A054535(d, v) = c_d(v) = Sum_{s | gcd(d,v)} s*Moebius(d/s). These are the Ramanujan sums, which equal also the von Sterneck function c_d(v) = phi(d) * Moebius(d/gcd(d, v))/phi(d/gcd(d, v)). We have A054535(d, v) = A054534(v, d).
It would be interesting to see whether there is a proof of the results by Fredman (1975), Elashvili et al. (1999), and Panyushev (2011) for a general v using Molien series as it is done by Sloane (2014) for the case v = 0 (in which case, A054535(d, 0) = phi(d)). (Even though the columns of array A054535(d, v) start at v = 1, we may start the array at column v = 0 as well.)
(End)
U(n, k) is the number of equivalence classes of k-tuples of residues modulo n, identifying those that differ componentwise by a constant and those that differ by a permutation. - Álvar Ibeas, Sep 21 2021

			Triangle starts:
[ 0]  1,
[ 1]  1,  1,
[ 2]  1,  1,  1,
[ 3]  1,  1,  1,  1,
[ 4]  1,  1,  2,  1,  1,
[ 5]  1,  1,  2,  2,  1,  1,
[ 6]  1,  1,  3,  4,  3,  1,  1,
[ 7]  1,  1,  3,  5,  5,  3,  1,  1,
[ 8]  1,  1,  4,  7, 10,  7,  4,  1,  1,
[ 9]  1,  1,  4, 10, 14, 14, 10,  4,  1,  1,
[10]  1,  1,  5, 12, 22, 26, 22, 12,  5,  1, 1,
[11]  1,  1,  5, 15, 30, 42, 42, 30, 15,  5, 1, 1,
[12]  1,  1,  6, 19, 43, 66, 80, 66, 43, 19, 6, 1, 1, ...

N. G. de Bruijn, Polya's theory of counting, in: Applied Combinatorial Mathematics (E. F. Beckenbach, ed.), John Wiley and Sons, New York, 1964, pp. 144-184 (implies g.f. for this triangle).
Richard Stanley, Enumerative Combinatorics, 2nd. ed., Vol 1, Chapter I, Problem 105, pp. 122 and 168, discusses the number of subsets of Z/nZ that add to 0. - N. J. A. Sloane, May 06 2014
J. Voß, Posting to Sequence Fans Mailing List, April 30, 2014.
H. S. Wilf, personal communication to N. J. A. Sloane, Nov., 1990.
See A000031 for many additional references and links.

Seiichi Manyama, Rows n=0..139 of triangle, flattened (Rows 0..50 from T. D. Noe)
Ethan Akin and Morton Davis, Bulgarian solitaire, Am. Math. Monthly 92 (4) (1985), 237-250.
J. Brandt, Cycles of partitions, Proc. Am. Math. Soc. 85 (3) (1982), 483-486, Theorem 5.
Paul Drube and Puttipong Pongtanapaisan, Annular Non-Crossing Matchings, Journal of Integer Sequences, Vol. 19 (2016), #16.2.4.
A. Elashvili and M. Jibladze, Hermite reciprocity for the regular representations of cyclic groups, Indag. Math. (N.S.) 9 (1998), no. 2, 233-238. MR1691428 (2000c:13006)
A. Elashvili, M. Jibladze, and D. Pataraia, Combinatorics of necklaces and "Hermite reciprocity", J. Algebraic Combin. 10 (1999), no. 2, 173--188. MR1719140 (2000j:05009). See p. 174. - _N. J. A. Sloane_, Aug 06 2014
M. L. Fredman, A symmetry relationship for a class of partitions, J. Combinatorial Theory Ser. A 18 (1975), 199-202.
Harold Fredricksen, An algorithm for generating necklaces of beads in two colors, Discrete Mathematics, Volume 61, Issues 2-3, September 1986, 181-188.
D. E. Knuth, Computer science and its relation to mathematics, Amer. Math. Monthly, 81 (1974), 323-343.
D. E. Knuth, H. Wilf, C. L. Mallows, and D. Klarner, Correspondence, 1994
Petr Lisonek, Computer-assisted Studies in Algebraic Combinatorics, Thesis, University of Linz (Austria) Sep. 1994, pp. 72-73.
D. I. Panyushev, Fredman's reciprocity, invariants of abelian groups, and the permanent of the Cayley table, J. Algebraic Combin. 33 (2011), 111-125.
Mónica A. Reyes, Cristina Dalfó, Miguel Àngel Fiol, and Arnau Messegué, A general method to find the spectrum and eigenspaces of the k-token of a cycle, and 2-token through continuous fractions, arXiv:2403.20148 [math.CO], 2024. See p. 5.
Frank Ruskey, Generate Necklaces, Lyndon words, and relatives, The Combinatorial Object Server.
Frank Ruskey and Joe Sawada, An Efficient Algorithm for Generating Necklaces with Fixed Density, SIAM J. Computing, 29 (1999) 671-684.
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
N. J. A. Sloane, A Note on Modular Partitions and Necklaces, 2014.
Pantelimon Stănică and Subhamoy Maitra, Rotation symmetric boolean functions - count and cryptographic properties, Disc. Appl. Math. 156 (2008) 1567-1580, g_{n,w} Theorem 9.
Wikipedia, Necklaces Animation [Broken link?]
Wolfram Research, Necklaces Applet.
Index entries for sequences related to necklaces

Row sums: A000031. Columns 0-12: A000012, A000012, A004526, A007997(n+5), A008610, A008646, A032191-A032197.
Cf. A051168, A052307, A052311-A052313.
See A037306 and A241926 for essentially identical triangles.
See A245558, A245559 for a closely related array.

A047996 := proc(n,k) local C,d; if k= 0 then return 1; end if; C := 0 ; for d in numtheory[divisors](igcd(n,k)) do C := C+numtheory[phi](d)*binomial(n/d,k/d) ; end do: C/n ; end proc:
seq(seq(A047996(n,k),k=0..n),n=0..10) ; # R. J. Mathar, Apr 14 2011

t[n_, k_] := Total[EulerPhi[#]*Binomial[n/#, k/#] & /@ Divisors[GCD[n, k]]]/n; t[0, 0] = 1; Flatten[Table[t[n, k], {n, 0, 13}, {k, 0, n}]] (* Jean-François Alcover, Jul 19 2011, after given formula *)

p(n) = if(n<=0, n==0, 1/n * sum(i=0,n-1, (x^(n/gcd(i,n))+1)^gcd(i,n) ));
for (n=0,17, print(Vec(p(n)))); /* print triangle */
/* Joerg Arndt, Sep 28 2012 */

T(n,k) = if(n<=0, n==0, 1/n * sumdiv(gcd(n,k), d, eulerphi(d)*binomial(n/d,k/d) ) );
/* print triangle: */
{ for (n=0, 17, for (k=0, n, print1(T(n,k),", "); ); print(); ); }
/* Joerg Arndt, Oct 21 2012 */

T(n, k) = (1/n) * Sum_{d | (n, k)} phi(d)*binomial(n/d, k/d).
T(2*n,n) = A003239(n); T(2*n+1,n) = A000108(n). - Philippe Deléham, Jul 25 2006
G.f. for row n (n>=1):  (1/n) * Sum_{i=0..n-1} ( x^(n/gcd(i,n)) + 1 )^gcd(i,n). - Joerg Arndt, Sep 28 2012
G.f.: Sum_{n, k >= 0} T(n, k)*x^n*y^k = 1 - Sum_{s>=1} (phi(s)/s)*log(1-x^s*(1+y^s)). - Petros Hadjicostas, Oct 26 2017
Product_{d >= 1} (1 - x^d - y^d) = Product_{i,j >= 0} (1 - x^i*y^j)^T(i+j, j), where not both i and j are zero. (It follows from Somos' infinite product for array A051168.) - Petros Hadjicostas, Jul 12 2019

Name edited by Petros Hadjicostas, Nov 16 2017

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

Original entry on oeis.org

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

A152176 Triangle read by rows: T(n,k) is the number of k-block partitions of an n-set up to rotations and reflections.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 3, 5, 2, 1, 1, 7, 14, 11, 3, 1, 1, 8, 31, 33, 16, 3, 1, 1, 17, 82, 137, 85, 27, 4, 1, 1, 22, 202, 478, 434, 171, 37, 4, 1, 1, 43, 538, 1851, 2271, 1249, 338, 54, 5, 1, 1, 62, 1401, 6845, 11530, 8389, 3056, 590, 70, 5, 1, 1, 121, 3838, 26148

Offset: 1

Vladeta Jovovic, Nov 27 2008

Number of bracelet structures of length n using exactly k different colored beads. Turning over will not create a new bracelet. Permuting the colors of the beads will not change the structure. - Andrew Howroyd, Apr 06 2017
The number of achiral structures (A) is given in A140735 (odd n) and A293181 (even n).  The number of achiral structures plus twice the number of chiral pairs (A+2C) is given in A152175.  These can be used to determine A+C by taking half their average, as is done in the Mathematica program. - Robert A. Russell, Feb 24 2018
T(n,k)=pi_k(C_n) which is the number of non-equivalent partitions of the cycle on n vertices, with exactly k parts. Two partitions P1 and P2 of a graph G are said to be equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. - Mohammad Hadi Shekarriz, Aug 21 2019

			Triangle begins:
  1;
  1,  1;
  1,  1,   1;
  1,  3,   2,    1;
  1,  3,   5,    2,    1;
  1,  7,  14,   11,    3,    1;
  1,  8,  31,   33,   16,    3,   1;
  1, 17,  82,  137,   85,   27,   4,  1;
  1, 22, 202,  478,  434,  171,  37,  4, 1;
  1, 43, 538, 1851, 2271, 1249, 338, 54, 5, 1;
  ...

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..1275
B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, arXiv:1910.12102 [math.CO], 2019.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
Tilman Piesk, Partition related number triangles
Marko Riedel, Bracelets with swappable colors classified by the distribution of colors, Power Group Enumeration algorithm
Marko Riedel, Maple code for the number of bracelets with some number of swappable colors by Power Group Enumeration
Mohammad Hadi Shekarriz, GAP Program

Columns 2-6 are A056357, A056358, A056359, A056360, A056361.
Row sums are A084708.
Partial row sums include A000011, A056353, A056354, A056355, A056356.
Cf. A081720, A273891, A008277 (set partitions), A284949 (up to reflection), A152175 (up to rotation).

Adn[d_, n_] := Adn[d, n] = Which[0==n, 1, 1==n, DivisorSum[d, x^# &],
  1==d, Sum[StirlingS2[n, k] x^k, {k, 0, n}],
  True, Expand[Adn[d, 1] Adn[d, n-1] + D[Adn[d, n - 1], x] x]];
Ach[n_, k_] := Ach[n, k] = Switch[k, 0, If[0==n, 1, 0], 1, If[n>0, 1, 0],
  (* else *) _, If[OddQ[n], Sum[Binomial[(n-1)/2, i] Ach[n-1-2i, k-1],
  {i, 0, (n-1)/2}], Sum[Binomial[n/2-1, i] (Ach[n-2-2i, k-1]
  + 2^i Ach[n-2-2i, k-2]), {i, 0, n/2-1}]]] (* achiral loops of length n, k colors *)
Table[(CoefficientList[DivisorSum[n, EulerPhi[#] Adn[#, n/#] &]/(x n), x]
+ Table[Ach[n, k],{k,1,n}])/2, {n, 1, 20}] // Flatten (* Robert A. Russell, Feb 24 2018 *)

\\ see A056391 for Polya enumeration functions
T(n,k) = NonequivalentStructsExactly(DihedralPerms(n), k); \\ Andrew Howroyd, Oct 14 2017

\\ Ach is A304972 and R is A152175 as square matrices.
Ach(n)={my(M=matrix(n, n, i, k, i>=k)); for(i=3, n, for(k=2, n, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}
R(n)={Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
T(n)={(R(n) + Ach(n))/2}
{ my(A=T(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Sep 20 2019

A005232 Expansion of (1-x+x^2)/((1-x)^2(1-x^2)(1-x^4)).

Original entry on oeis.org

1, 1, 3, 4, 8, 10, 16, 20, 29, 35, 47, 56, 72, 84, 104, 120, 145, 165, 195, 220, 256, 286, 328, 364, 413, 455, 511, 560, 624, 680, 752, 816, 897, 969, 1059, 1140, 1240, 1330, 1440, 1540, 1661, 1771, 1903, 2024, 2168, 2300, 2456, 2600, 2769, 2925, 3107, 3276

Offset: 0

N. J. A. Sloane

Also number of n X 2 binary matrices under row and column permutations and column complementations (if offset is 0).
Also Molien series for certain 4-D representation of dihedral group of order 8.
With offset 4, number of bracelets (turnover necklaces) of n-bead of 2 colors with 4 red beads. - Washington Bomfim, Aug 27 2008
From Vladimir Shevelev, Apr 23 2011: (Start)
Also number of non-equivalent necklaces of 4 beads each of them painted by one of n colors.
The sequence solves the so-called Reis problem about convex k-gons in case k=4 (see our comment to A032279). (End)
Number of 2 X 2 matrices with nonnegative integer values totaling n under row and column permutations. - Gabriel Burns, Nov 08 2016
From Petros Hadjicostas, Jan 12 2019: (Start)
By "necklace", Vladimir Shevelev (above) means "turnover necklace", i.e., a bracelet. Zagaglia Salvi (1999) also uses this terminology: she calls a bracelet "necklace" and a necklace "cycle".
According to Cyvin et al. (1997), the sequence (a(n): n >= 0) consists of "the total numbers of isomers of polycyclic conjugated hydrocarbons with q + 1 rings and q internal carbons in one ring (class Q_q)", where q = 4 and n is the hydrogen content (i.e., we count certain isomers of C_{n+2*q} H_n with q = 4 and n >= 0). (End)

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 8*x^4 + 10*x^5 + 16*x^6 + 20*x^7 + 29*x^8 + ...
There are 8 4 X 2 matrices up to row and column permutations and column complementations:
  [1 1] [1 0] [1 0] [0 1] [0 1] [0 1] [0 1] [0 0]
  [1 1] [1 1] [1 0] [1 0] [1 0] [1 0] [0 1] [0 1]
  [1 1] [1 1] [1 1] [1 1] [1 0] [1 0] [1 0] [1 0]
  [1 1] [1 1] [1 1] [1 1] [1 1] [1 0] [1 0] [1 1].
There are 8 2 X 2 matrices of nonnegative integers totaling 4 up to row and column permutations:
  [4 0] [3 1] [2 2] [2 1] [2 1] [3 0] [2 0] [1 1]
  [0 0] [0 0] [0 0] [0 1] [1 0] [1 0] [2 0] [1 1].

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. Zagaglia Salvi, Ordered partitions and colourings of cycles and necklaces, Bull. Inst. Combin. Appl., 27 (1999), 37-40.

T. D. Noe, Table of n, a(n) for n = 0..1000
Nesrine Benyahia-Tani, Zahra Yahi, and Sadek Bouroubi Ordered and non-ordered non-congruent convex quadrilaterals inscribed in a regular n-gon. Rostocker Math. Kolloq. 68, 71-79 (2013), Theorem 1.
S. J. Cyvin, B. N. Cyvin, J. Brunvoll, I. Gutman, Chen Rong-si, S. El-Basil, and Zhang Fuji, Polygonal systems including the corannulene and coronene homologs: novel applications of Pólya's theorem, Z. Naturforsch., 52a (1997), 867-873.
S. N. Ethier and S. E. Hodge, Identity-by-descent analysis of sibship configurations, Amer. J. Medical Genetics, 22 (1985), 263-272.
H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no. 8, 964-999.
W. D. Hoskins and Anne Penfold Street, Twills on a given number of harnesses, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15.
W. D. Hoskins and A. P. Street, Twills on a given number of harnesses, J. Austral. Math. Soc. (Series A), 33 (1982), 1-15. (Annotated scanned copy)
M. Klemm, Selbstduale Codes über dem Ring der ganzen Zahlen modulo 4, Arch. Math. (Basel), 53 (1989), 201-207.
P. Lisonek, Quasi-polynomials: A case study in experimental combinatorics, RISC-Linz Report Series No. 93-18, 1983. (Annotated scanned copy)
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992
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]
Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.
Vladimir Shevelev, A problem of enumeration of two-color bracelets with several variations, arXiv:0710.1370 [math.CO], 2007-2011.
Vladimir Shevelev, Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma) (Cf. Section 5), arXiv:1104.4051 [math.CO], 2011.
Index entries for sequences related to bracelets
Index entries for Molien series
Index entries for linear recurrences with constant coefficients, signature (2,0,-2,2,-2,0,2,-1).
Index entries for two-way infinite sequences

Row n=2 of A343875.
Column k=4 of A052307.
Cf. A006381, A006382, A008805.

A005232:=-(-1-z-2*z**3+2*z**2+z**7-2*z**6+2*z**4)/(z**2+1)/(1+z)**2/(z-1)**4; # conjectured by Simon Plouffe in his 1992 dissertation; gives sequence apart from an initial 1

k = 4; Table[(Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n + Binomial[If[OddQ[n], n - 1, n - If[OddQ[k], 2, 0]]/2, If[OddQ[k], k - 1, k]/2])/2, {n, k, 50}] (* Robert A. Russell, Sep 27 2004 *)
CoefficientList[ Series[(1 - x + x^2)/((1 - x)^2(1 - x^2)(1 - x^4)), {x, 0, 51}], x] (* Robert G. Wilson v, Mar 29 2006 *)
LinearRecurrence[{2,0,-2,2,-2,0,2,-1},{1,1,3,4,8,10,16,20},60] (* Harvey P. Dale, Oct 24 2012 *)
k=4 (* Number of red beads in bracelet problem *); CoefficientList[Series[(1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])+(1+x)/(1-x^2)^Floor[(k+2)/2])/2,{x,0,50}],x] (* Herbert Kociemba, Nov 04 2016 *)

{a(n) = (n^3 + 9*n^2 + (32-9*(n%2))*n + [48, 15, 36, 15][n%4+1]) / 48}; \\ Michael Somos, Feb 01 2007

{a(n) = my(s=1); if( n<-5, n = -6-n; s=-1); if( n<0, 0, s * polcoeff( (1 - x + x^2) / ((1 - x)^2 * (1 - x^2) * (1 - x^4)) + x * O(x^n), n))}; \\ Michael Somos, Feb 01 2007

a(n) = round((n^3 +9*n^2 +(32-9*(n%2))*n)/48 +0.6) \\ Washington Bomfim, Jul 17 2008

a(n) = ceil((n+1)*(2*n^2+16*n+39+9*(-1)^n)/96) \\ Tani Akinari, Aug 23 2013

a=lambda n: sum((k//2+1)*((n-k)//2+1) for k in range((n-1)//2+1))+(n+1)%2*(((n//4+1)*(n//4+2))//2)  # Gabriel Burns, Nov 08 2016

G.f.: (1+x^3)/((1-x)*(1-x^2)^2*(1-x^4)).
G.f.: (1/8)*(1/(1-x)^4+3/(1-x^2)^2+2/(1-x)^2/(1-x^2)+2/(1-x^4)). - Vladeta Jovovic, Aug 05 2000
Euler transform of length 6 sequence [ 1, 2, 1, 1, 0, -1 ]. - Michael Somos, Feb 01 2007
a(2n+1) = A006918(2n+2)/2;
a(2n) = (A006918(2n+1) + A008619(n))/2.
a(n) = -a(-6 - n) for all n in Z. - Michael Somos, Feb 05 2011
From Vladimir Shevelev, Apr 22 2011: (Start)
if n == 0 (mod 4), then a(n) = n*(n^2-3*n+8)/48;
if n == 1, 3 (mod 4), then a(n) = (n^2-1)*(n-3)/48;
if n == 2 (mod 4), then a(n) = (n-2)*(n^2-n+6)/48. (End)
a(n) = 2*a(n-1) - 2*a(n-3) + 2*a(n-4) - 2*a(n-5) + 2*a(n-7) - a(n-8) with a(0) = 1, a(1) = 1, a(2) = 3, a(3) = 4, a(4) = 8, a(5) = 10, a(6) = 16, a(7) = 20. - Harvey P. Dale, Oct 24 2012
a(n) = ((n+3)*(2*n^2+12*n+19+9*(-1)^n) + 6*(-1)^((2*n-1+(-1)^n)/4)*(1+(-1)^n))/96. - Luce ETIENNE, Mar 16 2015
a(n) = |A128498(n)| + |A128498(n-3)|. - R. J. Mathar, Jun 11 2019

Sequence extended by Christian G. Bower

A027671 Number of necklaces with n beads of 3 colors, allowing turning over.

Original entry on oeis.org

1, 3, 6, 10, 21, 39, 92, 198, 498, 1219, 3210, 8418, 22913, 62415, 173088, 481598, 1351983, 3808083, 10781954, 30615354, 87230157, 249144711, 713387076, 2046856566, 5884491500, 16946569371, 48883660146, 141217160458, 408519019449, 1183289542815

Offset: 0

Alford Arnold

Number of bracelets of n beads using up to three different colors. - Robert A. Russell, Sep 24 2018

			For n=2, the six bracelets are AA, AB, AC, BB, BC, and CC. - _Robert A. Russell_, Sep 24 2018

J. L. Fisher, Application-Oriented Algebra (1977), ISBN 0-7002-2504-8, circa p. 215.
M. Gardner, "New Mathematical Diversions from Scientific American" (Simon and Schuster, New York, 1966), pp. 245-246.

T. D. Noe, Table of n, a(n) for n = 0..200
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
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]
M. Taniguchi, H. Du, and J. S. Lindsey, Enumeration of virtual libraries of combinatorial modular macrocyclic (bracelet, necklace) architectures and their linear counterparts, Journal of Chemical Information and Modeling, 53 (2013), 2203-2216.
R. M. Thompson and R. T. Downs, Systematic generation of all nonequivalent closest-packed stacking sequences of length N using group theory, Acta Cryst. B57 (2001), 766-771; B58 (2002), 153.
Eric Weisstein's World of Mathematics, Necklace.
Index entries for sequences related to bracelets

Cf. A056353, A114438.
a(n) = A081720(n,3), n >= 3. - Wolfdieter Lang, Jun 03 2012
Column 3 of A051137.
a(n) = A278639(n) + A182751(n+1).
Equals A001867 - A278639.

Needs["Combinatorica`"];  Join[{1}, Table[CycleIndex[DihedralGroup[n], s]/.Table[s[i]->3, {i,1,n}], {n,1,30}]] (* Geoffrey Critzer, Sep 29 2012 *)
Needs["Combinatorica`"]; Join[{1}, Table[NumberOfNecklaces[n, 3, Dihedral], {n, 30}]] (* T. D. Noe, Oct 02 2012 *)
mx=40;CoefficientList[Series[(1-Sum[ EulerPhi[n]*Log[1-3*x^n]/n,{n,mx}]+(1+3 x+3 x^2)/(1-3 x^2))/2,{x,0,mx}],x] (* Herbert Kociemba, Nov 02 2016 *)
t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1+k)*k^(n/2))/(2*n), (t1 + n*k^((n+1)/2))/(2*n)]); a[0] = 1; a[n_] := t[n, 3]; Array[a, 30, 0] (* Jean-François Alcover, Nov 02 2017, after Maple code for A081720 *)
k=3; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/(2n) + (k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2])/4, {n, 1, 30}], 1] (* Robert A. Russell, Sep 24 2018 *)

a(n,k=3) = if(n==0,1,(k^floor((n+1)/2) + k^ceil((n+1)/2))/4 + (1/(2*n))* sumdiv(n, d, eulerphi(d)*k^(n/d) ) );
vector(55,n,a(n-1)) \\ Joerg Arndt, Oct 20 2019

G.f.: (1 - Sum_{n>=1} phi(n)*log(1 - 3*x^n)/n + (1+3*x+3*x^2)/(1-3*x^2))/2. - Herbert Kociemba, Nov 02 2016
For n > 0, a(n) = (k^floor((n+1)/2) + k^ceiling((n+1)/2))/4 + (1/(2*n))* Sum_{d|n} phi(d)*k^(n/d), where k=3 is the maximum number of colors. - Robert A. Russell, Sep 24 2018
a(0) = 1; a(n) = (k^floor((n+1)/2) + k^ceiling((n+1)/2))/4 + (1/(2*n))*Sum_{i=1..n} k^gcd(n,i), where k=3 is the maximum number of colors.
(See A075195 formulas.) - Richard L. Ollerton, May 04 2021
2*a(n) = A182751(n+1) + A001867(n), n>0.

More terms from Christian G. Bower

A005513 Number of n-bead bracelets (turnover necklaces) of two colors with 6 red beads and n-6 black beads.

Original entry on oeis.org

1, 1, 4, 7, 16, 26, 50, 76, 126, 185, 280, 392, 561, 756, 1032, 1353, 1782, 2277, 2920, 3652, 4576, 5626, 6916, 8372, 10133, 12103, 14448, 17063, 20128, 23528, 27474, 31824, 36822, 42315, 48564, 55404, 63133, 71554, 81004

Offset: 6

N. J. A. Sloane

From Vladimir Shevelev, Apr 23 2011: (Start)
Also number of non-equivalent (turnover) necklaces of 6 beads each of them painted by one of n colors.
The sequence solves the so-called Reis problem about convex k-gons in case k=6 (see our comment to A032279).
(End)
Also number of reverse invariant anonymous and neutral equivalence classes of preference profiles with 3 alternatives and (n-6) agents (IANC model). - Alexander Karpov, Apr 12 2018
Also the number of weighted cubic graphs with weight n derived from one of the 2 cubic graphs on 6 vertices (contributing to A321306). - R. J. Mathar, Nov 05 2018

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. Zagaglia Salvi, Ordered partitions and colourings of cycles and necklaces, Bull. Inst. Combin. Appl., 27 (1999), 37-40.

Vincenzo Librandi, Table of n, a(n) for n = 6..5000
S. J. Cyvin, B. N. Cyvin, J. Brunvoll, I. Gutman, Chen Rong-si, S. El-Basil, and Zhang Fuji, Polygonal systems including the corannulene and coronene homologs: novel applications of Pólya's theorem, Z. Naturforsch., 52a (1997), 867-873.
Hansraj Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no.8, 964-999.
W. D. Hoskins and Anne Penfold Street, Twills on a given number of harnesses, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15.
W. D. Hoskins and A. P. Street, Twills on a given number of harnesses, J. Austral. Math. Soc. (Series A), 33 (1982), 1-15. (Annotated scanned copy)
A. Karpov, An Informational Basis for Voting Rules, NRU Higher School of Economics. Series WP BRP "Economics/EC". 2018. No. 188
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]
Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.
Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.
Vladimir Shevelev, Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma), arXiv:1104.4051 [math.CO], 2011. (Cf. Section 5)
A. P. Street, Letter to N. J. A. Sloane, N.D.
Index entries for sequences related to bracelets
Index entries for linear recurrences with constant coefficients, signature (2,1,-3,-1,1,4,-3,-3,4,1,-1,-3,1,2,-1).

Column k=6 of A052307.

A005513 := proc(n) if n mod 6 = 0 then (24*binomial(n-1,5)+3*(n+1)*(n-2)*(n-4)+16*n)/288 elif n mod 6 = 3 then (24*binomial(n-1,5)+3*(n-1)*(n-3)*(n-5)+16*n-48)/288 elif n mod 6 = 2 or n mod 6 = 4 then (8*binomial(n-1,5)+(n+1)*(n-2)*(n-4))/96 elif n mod 6 = 1 or n mod 6 = 5 then (8*binomial(n-1,5)+(n-1)*(n-3)*(n-5))/96 fi: end: seq(A005513(n),n=6..44); # Johannes W. Meijer, Aug 11 2011

k = 6; Table[(Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n + Binomial[If[OddQ[n], n - 1, n - If[OddQ[k], 2, 0]]/2, If[OddQ[k], k - 1, k]/2])/2, {n, k, 50}] (* Robert A. Russell, Sep 27 2004 *)
k=6; CoefficientList[Series[x^k*(1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])+(1+x)/(1-x^2)^Floor[(k+2)/2])/2,{x,0,50}],x] (* Herbert Kociemba, Nov 04 2016 *)
CoefficientList[Series[(1/12) (1/(1 - x)^6 + 4/(1 - x^2)^3 + 2/(1 - x^3)^2 + 3/((1 - x)^2 (1 - x^2)^2) + 2/(1 - x^6)), {x, 0, 43}], x] (* Vincenzo Librandi, Apr 24 2018 *)

S. J. Cyvin et al. (1997) give a g.f.
G.f.: (x^6/12)*(1/(1-x)^6+4/(1-x^2)^3+2/(1-x^3)^2+3/((1-x)^2*(1-x^2)^2)+2/(1-x^6)). - Vladeta Jovovic, Feb 28 2007
G.f.: x^6*(1-x+x^2+x^3+2*x^4+2*x^6+x^8-x^5) / ( (x^2-x+1)*(1+x+x^2)^2*(1+x)^3*(x-1)^6 ). - R. J. Mathar, Sep 18 2011
From Vladimir Shevelev, Apr 23 2011: (Start)
if n==0 mod 6, a(n)=(24*C(n-1,5)+3*(n+1)*(n-2)*(n-4)+16*n)/288;
if n==3 mod 6, a(n)=(24*C(n-1,5)+3*(n-1)*(n-3)*(n-5)+16*n-48)/288;
if n==2,4 mod 6, a(n)=(8*C(n-1,5)+(n+1)*(n-2)*(n-4))/96;
if n==1,5 mod 6, a(n)=(8*C(n-1,5)+(n-1)*(n-3)*(n-5))/96.
(End)

Sequence extended and description corrected by Christian G. Bower

Name edited by Petros Hadjicostas, Jan 10 2019

A007123 Number of connected unit interval graphs with n nodes; also number of bracelets (turnover necklaces) with n black beads and n-1 white beads.

Original entry on oeis.org

1, 1, 2, 4, 10, 26, 76, 232, 750, 2494, 8524, 29624, 104468, 372308, 1338936, 4850640, 17685270, 64834550, 238843660, 883677784, 3282152588, 12233309868, 45741634536, 171530482864, 644953425740, 2430975800876, 9183681736376, 34766785487152, 131873995933480

Offset: 1

N. J. A. Sloane

Also number of rooted planar general trees (of n vertices or n-1 edges) up to reflection. - Antti Karttunen, Aug 09 2002 (For the correspondence with bracelets, start by considering Raney's lemma as explained by Graham, Knuth & Patashnik.)
Number of connected lattice path matroids on n elements up to isomorphism.
a(n) = number of noncrossing set partitions of [n] up to reflection (i<->n+1-i). Example: a(4) counts 123, 1-23, 13-2, 1-2-3 but not 12-3 because it is the reflection of 1-23. - David Callan, Oct 08 2005
From Vladimir Shevelev, Apr 23 2011: (Start)
Also number of non-equivalent necklaces of n beads, each of which is painted by one of 2*n-1 colors.
The sequence solves the so-called Reis problem about convex k-gons in case N=2*n-1, k=n. H. Gupta (1979) gave a full solution; I gave a short proof of Gupta's result and showed an equivalence of this problem and each of the following problems: the problem of enumerating the bracelets of n beads of 2 colors, k of them black, and the problem of enumerating the necklaces of k beads, each painted by one of n colors.
a(n) is an essentially unimprovable upper estimate for the number of distinct values of the permanent in (0,1)-circulants of order 2*n-1 with n 1's in every row. (End)
The number of Dyck paths of semilength n-1 up to reversal; that is, the number of Dyck paths of semilength n-1, treating as identical a path and that path when traveled in reverse order. - Noah A Rosenberg, Jan 28 2019

			x + x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 26*x^6 + 76*x^7 + 232*x^8 + 750*x^9 + ...

S. R. Finch, Mathematical Constants, Cambridge, 2003, Section 5.6.7.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 345 & 346.
R. W. Robinson, personal communication.
R. W. Robinson, Numerical implementation of graph counting algorithms, AGRC Grant, Math. Dept., Univ. Newcastle, Australia, 1980.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1670 (first 190 terms from R. W. Robinson)
J. E. Bonin, A. de Mier, and M. Noy, Lattice path matroids: enumerative aspects and Tutte polynomials, arXiv:math/0211188 [math.CO], 2002.
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs., Vol. 3 (2000), #00.1.5.
Hansraj Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no. 8, 964-999.
Z. M. Himwich and N. A. Rosenberg, Roadblocked monotonic paths and the enumeration of coalescent histories for non-matching caterpillar gene trees and species trees, arXiv:1901.04465 [qbio.PE], 2019; Adv. Appl. Math. 113 (2020), 101939. (cf. Table 1)
Claudio Procesi, Some special bases of the 2-swap algebras, arXiv:2406.18687 [math.QA], 2024. See p. 3.
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]
Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.
Vladimir Shevelev, Spectrum of permanent's values and its extremal magnitudes in Lambda_n^3 and Lambda_n(alpha,beta,gamma), arXiv:1104.4051 [math.CO], 2011. (Cf. Section 5).
Index entries for sequences related to bracelets

Cf. A000108, A002420, A007595, A063886, A073201.
Occurs as row 164 in A073201.
Next-to-center columns of triangle A052307.
Equal to A001405 plus A006079.

f[k_Integer, n_] := (Plus @@ (EulerPhi[ # ]Binomial[n/#, k/# ] & /@ Divisors[GCD[n, k]])/n + Binomial[(n - If[OddQ@n, 1, If[OddQ@k, 2, 0]])/2, (k - If[OddQ@k, 1, 0])/2])/2 (* Robert A. Russell, Sep 27 2004 *)
Table[ f[n, 2n - 1], {n, 10}]
(* Comment from Wouter Meeussen, Feb 02 2013, added by N. J. A. Sloane, Feb 02 2013: To get lists of the necklaces in Mathematica, use (if n=4, say):
<

{a(n) = if( n<1, 0, (2 * binomial(n-1, (n-1)\2) + binomial(2*n, n) / (2*n - 1)) / 4)} /* Michael Somos, Apr 16 2012 */

from sympy import catalan, binomial, floor
def a(n): return 1 if n==1 else (catalan(n - 1) + binomial(n - 1, floor((n - 1)/2)))/2 # Indranil Ghosh, Jun 03 2017

a(n+1) = (Catalan(n) + binomial(n, floor(n/2)))/2 = (A000108(n) + A001405(n))/2. - Antti Karttunen, Aug 09 2002
G.f.: (1 + 2*x - sqrt(1 - 4*x)*sqrt(1 - 4*x^2))/(4*sqrt(1 - 4*x^2)).
G.f.: (sqrt((1 + 2*x) / (1 - 2*x)) - sqrt(1 - 4*x)) / 4. - Michael Somos, Apr 16 2012
a(n) = (A063886(n) - A002420(n)) / 4. - Michael Somos, Apr 16 2012
D-finite with recurrence n*(n-1)*(n-4)*a(n) - 4*(n-1)*(n^2-5*n+5)*a(n-1) - 4*(n-2)*(n^2-7*n+11)*a(n-2) + 8*(2*n-7)*(n-2)*(n-3)*a(n-3)=0. - R. J. Mathar, Aug 22 2018

Extended by Christian G. Bower

Edited by Jon E. Schoenfield, Feb 14 2015

cp's OEIS Frontend

A019536 Number of length n necklaces with integer entries that cover an initial interval of positive integers.

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A000013 Definition (1): Number of n-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed.

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A000029 Number of necklaces with n beads of 2 colors, allowing turning over (these are also called bracelets).

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A047996 Triangle read by rows: T(n,k) is the (n,k)-th circular binomial coefficient, where 0 <= k <= n.

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

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A152176 Triangle read by rows: T(n,k) is the number of k-block partitions of an n-set up to rotations and reflections.

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A005232 Expansion of (1-x+x^2)/((1-x)^2(1-x^2)(1-x^4)).