A051137 -id:A051137 - cp's OEIS frontend

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

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

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

A032275 Number of bracelets (turnover necklaces) of n beads of 4 colors.

Original entry on oeis.org

4, 10, 20, 55, 136, 430, 1300, 4435, 15084, 53764, 192700, 704370, 2589304, 9608050, 35824240, 134301715, 505421344, 1909209550, 7234153420, 27489127708, 104717491064, 399827748310, 1529763696820

Offset: 1

Christian G. Bower

nonn

			For n=2, the ten bracelets are AA, AB, AC, AD, BB, BC, BD, CC, CD, and DD. - _Robert A. Russell_, Sep 24 2018

C. G. Bower, Transforms (2)
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.
Index entries for sequences related to bracelets

Column 4 of A051137.

mx=40;CoefficientList[Series[(1-Sum[ EulerPhi[n]*Log[1-4*x^n]/n,{n,mx}]+(1+4 x+6 x^2)/(1-4 x^2))/2,{x,0,mx}],x] (* Herbert Kociemba, Nov 02 2016 *)
k=4; Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/(2n) + (k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2])/4, {n, 1, 30}] (* Robert A. Russell, Sep 24 2018 *)

"DIK" (bracelet, indistinct, unlabeled) transform of 4, 0, 0, 0, ...
Equals (A001868(n) + A056486(n)) / 2 = A001868(n) - A278640(n) = A278640(n) + A056486(n), for n>=1.
a(n) = A081720(n,4), n >= 4. - Wolfdieter Lang, Jun 03 2012
G.f.: (1 - Sum_{n>=1} phi(n)*log(1 - 4*x^n)/n + (1+4*x+6*x^2)/(1-4*x^2))/2. - Herbert Kociemba, Nov 02 2016
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=4 is the maximum number of colors. - Robert A. Russell, Sep 24 2018
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=4 is the maximum number of colors. (See A075195 formulas.) - Richard L. Ollerton, May 04 2021

A032276 Number of bracelets (turnover necklaces) with n beads of 5 colors.

Original entry on oeis.org

5, 15, 35, 120, 377, 1505, 5895, 25395, 110085, 493131, 2227275, 10196680, 46989185, 218102685, 1017448143, 4768969770, 22440372245, 105966797755, 501938733555, 2384200683816, 11353290089305

Offset: 1

Christian G. Bower

nonn

From Petros Hadjicostas, Sep 01 2018: (Start)
The DIK transform of the sequence (c(n): n >= 1), with g.f. C(x) = Sum_{n >= 1} c(n)*x^n, has g.f. -(1/2)*Sum_{m >= 1} (phi(m)/m))*log(1-C(x^m)) + (1 + C(x))^2/(4*(1-C(x^2))) - 1/4.
Here, c(1) = 5 and c(n) = 0 for n >= 2, and thus, C(x) = 5*x. Substituting this to the above g.f., we get that the g.f. of the current sequence is A(x) = Sum_{n >= 1} a(n)*x^n = -(1/2)*Sum_{m >= 1} (phi(m)/m))*log(1-5*x^m) + (1 + 5*x)^2/(4*(1-5*x^2)) - 1/4. This agrees with Herbert Kociemba's g.f. below except for an extra 1 because (1 + (1+5*x+10*x^2)/(1-5*x^2))/2 = 1 + (1 + 5*x)^2/(4*(1-5*x^2)) - 1/4.
(End)

			For n=2, the 15 bracelets are AA, AB, AC, AD, AE, BB, BC, BD, BE, CC, CD, CE, DD, DE, and EE. - _Robert A. Russell_, Sep 24 2018

C. G. Bower, Transforms (2)
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.
Index entries for sequences related to bracelets

Cf. A081720.
Column 5 of A051137.
Cf. A001869 (oriented), A056487 (achiral), A278641 (chiral).

mx=40;CoefficientList[Series[(1-Sum[ EulerPhi[n]*Log[1-5*x^n]/n,{n,mx}]+(1+5 x+10 x^2)/(1-5 x^2))/2,{x,0,mx}],x] (* Herbert Kociemba, Nov 02 2016 *)
k=5; Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/(2n) + (k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2])/4, {n, 1, 30}] (* Robert A. Russell, Sep 24 2018 *)

"DIK" (bracelet, indistinct, unlabeled) transform of 5, 0, 0, 0, ...
a(n) = A081720(n,5), n >= 1. - Wolfdieter Lang, Jun 03 2012
G.f.: (1 - Sum_{n>=1} phi(n)*log(1 - 5*x^n)/n + (1+5*x+10*x^2)/(1-5*x^2))/2. - Herbert Kociemba, Nov 02 2016
a(n) = (3/2)*5^(n/2) + (1/(2*n))*Sum_{d|n} phi(n/d)*5^d, if n is even, and = (1/2)*5^((n+1)/2) + (1/(2*n))*Sum_{d|n} phi(n/d)*5^d, if n is odd. - Petros Hadjicostas, Sep 01 2018
a(n) = (A001869(n) + A056487(n+1)) / 2 = A278641(n) + A056487(n+1) = A001869(n) - A278641(n). - Robert A. Russell, Oct 13 2018
a(n) = (k^floor((n+1)/2) + k^ceiling((n+1)/2))/4 + (1/(2*n))*Sum_{d divides n} phi(d)*k^(n/d), where k=5 is the maximum number of colors. - Richard L. Ollerton, May 04 2021
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=5 is the maximum number of colors. (See A051137.) - Richard L. Ollerton, May 04 2021

A056341 Number of bracelets of length n using a maximum of six different colored beads.

Original entry on oeis.org

6, 21, 56, 231, 888, 4291, 20646, 107331, 563786, 3037314, 16514106, 90782986, 502474356, 2799220041, 15673673176, 88162676511, 497847963696, 2821127825971, 16035812864946, 91404068329560

Offset: 1

Marks R. Nester

nonn

Turning over will not create a new bracelet.

			For n=2, the 21 bracelets are AA, AB, AC, AD, AE, AF, BB, BC, BD, BE, BF, CC, CD, CE, CF, DD, DE, DF, EE, EF, and FF. - _Robert A. Russell_, Sep 24 2018

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.]

Cf. A000029, A054625.
Cf. a(n) = A081720(n,6), n >= 6. - Wolfdieter Lang, Jun 03 2012
Column 6 of A051137.
Equals (A054625 + A056488) / 2 = A054625 - A278642 = A278642 + A056488.

mx=40;CoefficientList[Series[(1-Sum[ EulerPhi[n]*Log[1-6*x^n]/n,{n,mx}]+(1+6 x+15 x^2)/(1-6 x^2))/2,{x,0,mx}],x] (* Herbert Kociemba, Nov 02 2016 *)
k=6; Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/(2n) + (k^Floor[(n+1)/2] + k^Ceiling[(n+1)/2])/4, {n, 1, 30}] (* Robert A. Russell, Sep 24 2018 *)

a(n) = Sum_{d|n} phi(d)*6^(n/d)/(2*n);
a(n) = 6^((n+1)/2)/2 for n odd,
       (7/4)*6^(n/2) for n even.
G.f.: (1 - Sum_{n>=1} phi(n)*log(1 - 6*x^n)/n + (1+6*x+15*x^2)/(1-6*x^2))/2. - Herbert Kociemba, Nov 02 2016

A321791 Table read by descending antidiagonals: T(n,k) is the number of unoriented cycles (bracelets) of length n using up to k available colors.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 3, 3, 1, 0, 1, 4, 6, 4, 1, 0, 1, 5, 10, 10, 6, 1, 0, 1, 6, 15, 20, 21, 8, 1, 0, 1, 7, 21, 35, 55, 39, 13, 1, 0, 1, 8, 28, 56, 120, 136, 92, 18, 1, 0, 1, 9, 36, 84, 231, 377, 430, 198, 30, 1, 0

Offset: 0

Robert A. Russell, Dec 18 2018

			Table begins with T(0,0):
  1 1  1    1     1      1       1        1        1         1         1 ...
  0 1  2    3     4      5       6        7        8         9        10 ...
  0 1  3    6    10     15      21       28       36        45        55 ...
  0 1  4   10    20     35      56       84      120       165       220 ...
  0 1  6   21    55    120     231      406      666      1035      1540 ...
  0 1  8   39   136    377     888     1855     3536      6273     10504 ...
  0 1 13   92   430   1505    4291    10528    23052     46185     86185 ...
  0 1 18  198  1300   5895   20646    60028   151848    344925    719290 ...
  0 1 30  498  4435  25395  107331   365260  1058058   2707245   6278140 ...
  0 1 46 1219 15084 110085  563786  2250311  7472984  21552969  55605670 ...
  0 1 78 3210 53764 493131 3037314 14158228 53762472 174489813 500280022 ...
For T(3,3)=10, the unoriented cycles are 9 achiral (AAA, AAB, AAC, ABB, ACC, BBB, BBC, BCC, CCC) and 1 chiral pair (ABC-ACB).

Index entries for sequences related to bracelets

Cf. A075195 (oriented), A293496(chiral), A284855 (achiral).
Cf. A051137 (ascending antidiagonals).
Columns 0-6 are A000007, A000012, A000029, A027671, A032275, A032276, and A056341.
Rows 0-7 are A000012, A001477, A000217, A000292, A002817, A060446, A027670, and A060532.
Main diagonal gives A081721.

Table[If[k>0, DivisorSum[k, EulerPhi[#](n-k)^(k/#)&]/(2k) + ((n-k)^Floor[(k+1)/2]+(n-k)^Ceiling[(k+1)/2])/4, 1], {n, 0, 12}, {k, 0, n}] // Flatten

T(n,k) = [n==0] + [n>0] * (k^floor((n+1)/2) + k^ceiling((n+1)/2)) / 4 + (1/(2*n)) * Sum_{d|n} phi(d) * k^(n/d).
T(n,k) = (A075195(n,k) + A284855(n,k)) / 2.
T(n,k) = A075195(n,k) - A293496(n,k) = A293496(n,k) + A284855(n,k).
Linear recurrence for row n: T(n,k) = Sum_{j=0..n} -binomial(j-n-1,j+1) * T(n,k-1-j) for k >= n + 1.
O.g.f. for column k >= 0: Sum_{n>=0} T(n,k)*x^n = 3/4 + (1 + k*x)^2/(4*(1 - k*x^2)) - (1/2) * Sum_{d >= 1} (phi(d)/d) * log(1 - k*x^d). - Petros Hadjicostas, Feb 07 2021

cp's OEIS Frontend

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

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

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A032275 Number of bracelets (turnover necklaces) of n beads of 4 colors.

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A032276 Number of bracelets (turnover necklaces) with n beads of 5 colors.

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A056341 Number of bracelets of length n using a maximum of six different colored beads.

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A321791 Table read by descending antidiagonals: T(n,k) is the number of unoriented cycles (bracelets) of length n using up to k available colors.

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