A000011 -id:A000011 - cp's OEIS frontend

A032242 Number of identity bracelets of n beads of 5 colors.

5, 10, 10, 45, 252, 1120, 5270, 23475, 106950, 483504, 2211650, 10148630, 46911060, 217863040, 1017057256, 4767774375, 22438419120, 105960830300, 501928967930, 2384170903140, 11353241255900

Offset: 1

Christian G. Bower

nonn

For n>2 also number of asymmetric bracelets with n beads of five colors. - Herbert Kociemba, Nov 29 2016

Robert Israel, Table of n, a(n) for n = 1..1434
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]
Index entries for sequences related to bracelets

Column k=5 of A309528 for n >= 3.

N:= 50: # for a(1)..a(N)
G:= add(1/2*numtheory:-mobius(n)*(-log(1-5*x^n)/n - add(binomial(5,i)*x^(n*i)/(1-5*x^(2*n)),i=0..2)), n=1..N):
S:= series(G,x,N+1):
5,10,seq(coeff(S,x,j),j=3..N); # Robert Israel, Jun 24 2019

m=5; (* asymmetric bracelets of n beads of m colors *) Table[Sum[MoebiusMu[d](m^(n/d)/n - If[OddQ[n/d], m^((n/d+1)/2), ((m+1)m^(n/(2d))/2)]), {d,Divisors[n]}]/2, {n,3,20}] (* Robert A. Russell, Mar 18 2013 *)
mx=40;gf[x_,k_]:=Sum[MoebiusMu[n]*(-Log[1-k*x^n]/n-Sum[Binomial[k,i]x^(n i),{i,0,2}]/(1-k x^(2n)))/2,{n,mx}];ReplacePart[Rest[CoefficientList[Series[gf[x,5],{x,0,mx}],x]],{1->5,2->10}] (* Herbert Kociemba, Nov 29 2016 *)

a(n)={if(n<3, binomial(5,n), sumdiv(n, d, moebius(n/d)*(5^d/n - if(d%2, 5^((d+1)/2), 3*5^(d/2))))/2)} \\ Andrew Howroyd, Sep 12 2019

"DHK" (bracelet, identity, unlabeled) transform of 5, 0, 0, 0...
From Herbert Kociemba, Nov 29 2016: (Start)
More generally, gf(k) is the g.f. for the number of asymmetric bracelets with n beads of k colors.
gf(k): Sum_{n>=1} mu(n)*( -log(1-k*x^n)/n - Sum_{i=0..2} binomial(k,i)x^(n*i)/(1-k*x^(2*n)) )/2. (End)

A136704 Number of Lyndon words on {1,2,3} with an odd number of 1's and an odd number of 2's.

Original entry on oeis.org

0, 1, 2, 5, 12, 30, 78, 205, 546, 1476, 4026, 11070, 30660, 85410, 239144, 672605, 1899120, 5380830, 15292914, 43584804, 124527988, 356602950, 1023295422, 2941974270, 8472886092, 24441017580, 70607383938

Offset: 1

Jennifer Woodcock (jennifer.woodcock(AT)ugdsb.on.ca), Jan 16 2008

nonn

This sequence is also the number of Lyndon words on {1,2,3} with an even number of 1's and an odd number of 2's except that a(1) = 1 in this case.

Also, a Lyndon word is the aperiodic necklace representative which is lexicographically earliest among its cyclic shifts. Thus we can apply the fixed density formulas: L_k(n,d) = Sum L(n-d, n_1,..., n_(k-1)); n_1 + ... +n_(k-1) = d where L(n_0, n_1,...,n_(k-1)) = (1/n) Sum mu(j)*[(n/j)!/((n_0/j)!(n_1/j)!...(n_(k-1)/j)!)]; j|gcd(n_0, n_1,...,n_(k-1)). For this sequence, sum over n_0, n_1 = odd.

			For n = 3, out of 8 possible Lyndon words: 112, 113, 122, 123, 132, 133, 223, 233, only 123 and 132 have an odd number of both 1's and 2's. Thus a(3) = 2.

M. Lothaire, Combinatorics on Words, Addison-Wesley, Reading, MA, 1983.

Amiram Eldar, Table of n, a(n) for n = 1..1000
E. N. Gilbert and John Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
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]
Frank Ruskey and Joe Sawada, An Efficient Algorithm for Generating Necklaces with Fixed Density, SIAM J. Computing, 29 (1999), 671-684.
Mike Zabrocki, MATH5020 York University Course Website.

Cf. A006575, A027376, A133267, A136703.

Bisections: A253076, A253077.

a[1] = 0;
a[n_] := If[OddQ[n], Sum[MoebiusMu[d] * 3^(n/d), {d, Divisors[n]}], Sum[Boole[OddQ[d]] MoebiusMu[d] * (3^(n/d)-1), {d, Divisors[n]}]]/(4n);
Array[a, 27] (* Jean-François Alcover, Aug 26 2019 *)

a(n) = if (n==1, 0, if (n % 2, sumdiv(n, d, moebius(d)*3^(n/d))/(4*n), sumdiv(n, d, if (d%2, moebius(d)*(3^(n/d)-1)))/(4*n))); \\ Michel Marcus, Aug 26 2019

a(1) = 0; for n>1, if n = odd then a(n) = Sum_{d|n} (mu(d)*3^(n/d))/(4n). If n = even, then a(n) = Sum_{d|n, d odd} (mu(d)*(3^(n/d)-1))/(4n).

A261531 Number of necklaces with n beads of unlabeled colors such that the numbers of beads per color are distinct.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 15, 25, 69, 254, 1799, 4039, 16828, 61751, 349831, 3485031, 10391139, 49433136, 240065255, 1282012987, 9167583734, 131550812011, 459677216341, 2707382738559, 14318807603110, 94084166753927, 601900541251447, 5894253303715375

Offset: 0

Alois P. Heinz, Aug 23 2015

nonn

			a(4) = 2: 0000, 0001.
a(5) = 4: 00000, 00001, 00011, 00101.
a(6) = 15: 000000, 000001, 000011, 000101, 000112, 000121, 000122, 001001, 001012, 001021, 001022, 001102, 001201, 001202, 010102.

Alois P. Heinz, Table of n, a(n) for n = 0..260
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, Necklace
Wikipedia, Necklace (combinatorics)
Index entries for sequences related to necklaces

Cf. A072605, A261599, A261600.

with(numtheory): with(combinat):
g:= l-> (n-> `if`(n=0, 1, add(phi(j)*multinomial(n/j,
        (l/j)[]), j=divisors(igcd(l[])))/n))(add(i, i=l)):
b:= proc(n, i, l) `if`(i*(i+1)/2n, 0, b(n-i, i-1, [l[], i]))))
    end:
a:= n-> b(n$2, []):
seq(a(n), n=0..35);

multinomial[n_, k_] := n!/Times @@ (k!);
g[l_] := Function[n, If[n==0, 1, Sum[EulerPhi[j]*multinomial[n/j, l/j], {j, Divisors[GCD @@ l]}]/n]][Total[l]];
b[n_, i_, l_] := If[i*(i+1)/2n, 0, b[n-i, i-1, Append[l, i]]]]];
a[n_] := b[n, n, {}];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Mar 21 2017, translated from Maple *)

a(n)={if(n==0, 1, my(p=prod(k=1, n, (1+x^k/k!) + O(x*x^n))); sumdiv(n, d, eulerphi(n/d)*d!*polcoeff(p, d))/n)} \\ Andrew Howroyd, Dec 21 2017

a(n) = (1/n) * Sum_{d | n} phi(n/d) * A007837(d) for n>0. - Andrew Howroyd, Apr 02 2017

A261599 Number of primitive (aperiodic, or Lyndon) necklaces with n beads of unlabeled colors such that the numbers of beads per color are distinct.

Original entry on oeis.org

1, 1, 0, 1, 1, 3, 13, 24, 67, 252, 1795, 4038, 16812, 61750, 349806, 3485026, 10391070, 49433135, 240064988, 1282012986, 9167581934, 131550811985, 459677212302, 2707382738558, 14318807586215, 94084166753923, 601900541189696, 5894253303715121

Offset: 0

Alois P. Heinz, Aug 25 2015

nonn

			a(4) = 1: 0001.
a(5) = 3: 00001, 00011, 00101.
a(6) = 13: 000001, 000011, 000101, 000112, 000121, 000122, 001012, 001021, 001022, 001102, 001201, 001202, 010102.
a(7) = 24: 0000001, 0000011, 0000101, 0000111, 0000112, 0000121, 0000122, 0001001, 0001011, 0001012, 0001021, 0001022, 0001101, 0001102, 0001201, 0001202, 0010011, 0010012, 0010021, 0010022, 0010101, 0010102, 0010201, 0010202.

Alois P. Heinz, Table of n, a(n) for n = 0..300
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, Necklace
Wikipedia, Lyndon word
Wikipedia, Necklace (combinatorics)
Index entries for sequences related to necklaces

Cf. A007837, A261531, A261600.

with(numtheory):
b:= proc(n, i, g, d, j) option remember; `if`(i*(i+1)/20
       and gn, 0, binomial(n/j, i/j)*b(n-i, i-1, igcd(i, g), d, j))))
    end:
a:= n-> `if`(n=0, 1, add(add((f-> `if`(f=0, 0, f*b(n$2, 0, d, j)))(
                     mobius(j)), j=divisors(d)), d=divisors(n))/n):
seq(a(n), n=0..30);

a[0] = 1; a[n_] := With[{P = Product[1 + x^k/k!, {k, 1, n}] + O[x]^(n+1) // Normal}, DivisorSum[n, MoebiusMu[n/#]*#!*Coefficient[P, x, #]&]/n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 28 2018, after Andrew Howroyd *)

a(n)={if(n==0, 1, my(p=prod(k=1, n, (1+x^k/k!) + O(x*x^n))); sumdiv(n, d, moebius(n/d)*d!*polcoeff(p, d))/n)} \\ Andrew Howroyd, Dec 21 2017

a(n) = (1/n) * Sum_{d | n} moebius(n/d) * A007837(d) for n>0. - Andrew Howroyd, Dec 21 2017

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

A005516 Number of n-bead bracelets (turnover necklaces) with 12 red beads.

Original entry on oeis.org

1, 1, 7, 19, 72, 196, 561, 1368, 3260, 7105, 14938, 29624, 56822, 104468, 186616, 322786, 544802, 896259, 1444147, 2278640, 3532144, 5380034, 8070400, 11926928, 17393969, 25042836, 35638596, 50152013, 69855536

Offset: 12

N. J. A. Sloane

nonn

From Vladimir Shevelev, Apr 23 2011: (Start)
Also number of non-equivalent (turnover) necklaces of 12 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=12 (see our comment to A032279). (End)

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

Andrew Howroyd, Table of n, a(n) for n = 12..1000
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)
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]
V. Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35 (2004), no. 5, 629-638.
V. 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 (4,-4,-2,4,-4,12,-12,2,2,-12,24,-18,4,4,-6,15,-20,0,10,-4,10,0,-20,15,-6,4,4,-18,24,-12,2,2,-12,12,-4,4,-2,-4,4,-1).

Column k=12 of A052307.

k = 12; 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=12;CoefficientList[Series[x^k*(1/k Plus@@(EulerPhi[#] (1-x^#)^(-(k/#))&/@Divisors[k])+(1+x)/(1-x^2)^Floor[k/2+1])/2,{x,0,50}],x] (* Herbert Kociemba, Nov 04 2016 *)

Let s(n,k,d) = 1, if n==k (mod d), s(n,k,d) = 0, otherwise. Then a(n) = s(n,0,12)/6 + (n-6)*s(n,0,6)/72 + (n-4)*(n-8)*s(n,0,4)/384 + (n-3)*(n-6)*(n-9)*s(n,0,3)/1944 + (3840*C(n-1,11) + (n+1)*(n-2)*(n-4)*(n-6)*(n-8)*(n-10))/92160, if n is even; a(n) = (n-3)*(n-6)*(n-9)*s(n,0,3)/1944 + (3840*C(n-1,11) + (n-1)*(n-3)*(n-5)*(n-7)*(n-9)*(n-11))/92160, if n is odd. - Vladimir Shevelev, Apr 23 2011
From Herbert Kociemba, Nov 04 2016: (Start)
G.f.: 1/2*x^12*((1+x)/(1-x^2)^7 + 1/12*(1/(-1+x)^12 + 1/(-1+x^2)^6 + 2/(-1+x^3)^4 - 2/(-1+x^4)^3 + 2/(-1+x^6)^2 - 4/(-1+x^12))).
G.f.: k=12, x^k*((1/k)*(Sum_{d|k} phi(d)*(1 - x^d)^(-k/d)) + (1 + x)/(1 -x^2)^floor((k+2)/2))/2. (End)

Sequence extended and description corrected by Christian G. Bower

A005654 Number of bracelets (turn over necklaces) with n red, 1 pink and n-1 blue beads; also reversible strings with n red and n-1 blue beads; also next-to-central column in Losanitsch's triangle A034851.

Original entry on oeis.org

1, 2, 6, 19, 66, 236, 868, 3235, 12190, 46252, 176484, 676270, 2600612, 10030008, 38781096, 150273315, 583407990, 2268795980, 8836340260, 34461678394, 134564560988, 526024917288, 2058358034616, 8061901596814, 31602652961516, 123979635837176, 486734861612328

Offset: 1

N. J. A. Sloane

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

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Marcia Ascher, Mu torere: an analysis of a Maori game, Math. Mag. 60 (1987), no. 2, 90-100.
R. K. Guy & N. J. A. Sloane, Correspondence, 1985
A. Ivanyi, L. Lucz, T. Matuszka, and S. Pirzada, Parallel enumeration of degree sequences of simple graphs, Acta Univ. Sapientiae, Informatica, 4, 2 (2012) 260-288.
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]
N. J. A. Sloane, Classic Sequences
Index entries for sequences related to bracelets

Cf. A034851.

[((Binomial(2*n-1, n)+Binomial(n-1, Floor(n/2)))/2): n in [1..30]]; // Vincenzo Librandi, May 24 2012

A005654:=n->(1/2)*(binomial(2*n-1,n)+binomial(n-1,floor(n/2))): seq(A005654(n), n=1..40); # Wesley Ivan Hurt, Jan 29 2017

Table[(Binomial[2n-1,n]+Binomial[n-1,Floor[n/2]])/2,{n,30}] (* Harvey P. Dale, May 17 2012 *)

C(n,k)=binomial(n,k)
a(n)=(1/2)*(C(2*n-1,n)+C(n-1,n\2))

a(n) = (1/2) * (binomial(2*n-1, n) + binomial(n-1, floor(n/2))). - Michael Somos
a(n) = A034851(2*n-1, n-1).
Conjecture: n*(n-2)*a(n) - (5*n-3)*(n-2)*a(n-1) + 4*(n-2)*a(n-2) + 4*(5*n^2-27*n+37)*a(n-3) - 8*(2*n-7)*(n-4)*a(n-4) = 0. - R. J. Mathar, Nov 09 2013

Sequence extended and description corrected by Christian G. Bower

A032165 Number of aperiodic necklaces of n beads of 10 colors.

Original entry on oeis.org

10, 45, 330, 2475, 19998, 166485, 1428570, 12498750, 111111000, 999989991, 9090909090, 83333249175, 769230769230, 7142856428565, 66666666659934, 624999993750000, 5882352941176470, 55555555499944500

Offset: 1

Christian G. Bower

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..500
C. G. Bower, Transforms (2)
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
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]
Index entries for sequences related to Lyndon words

Column 10 of A074650.

f[d_]:=MoebiusMu[d] 10^(n/d)/n; a[n_]:=Total[f/@Divisors[n]]; a[0]=1; Table[a[n], {n, 1, 20}] (* Vincenzo Librandi, Oct 14 2017 *)

a(n) = sumdiv(n, d, moebius(d)*10^(n/d))/n; \\ Andrew Howroyd, Oct 13 2017

"CHK" (necklace, identity, unlabeled) transform of 10, 0, 0, 0...
a(n) = Sum_{d|n} mu(d)*10^(n/d)/n.
G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 10*x^k))/k. - Ilya Gutkovskiy, May 19 2019

A032166 Number of aperiodic necklaces of n beads of 11 colors.

Original entry on oeis.org

11, 55, 440, 3630, 32208, 295020, 2783880, 26793030, 261994040, 2593726344, 25937424600, 261535549220, 2655593241840, 27124986721140, 278483211283552, 2871858103075830, 29732178147017280

Offset: 1

Christian G. Bower

nonn

Number of monic irreducible polynomials of degree n over GF(11). # Robert Israel, Jan 07 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..500
C. G. Bower, Transforms (2)
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
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]
Index entries for sequences related to Lyndon words

Column 11 of A074650.

f:= (n,p) -> add(numtheory:-mobius(d)*p^(n/d),d=numtheory:-divisors(n))/n:
seq(f(n,11), n=1..100); # Robert Israel, Jan 07 2015

f[d_]:=MoebiusMu[d] 11^(n/d)/n; a[n_]:=Total[f/@Divisors[n]]; a[0]=1; Table[a[n], {n, 1, 30}] (* Vincenzo Librandi, Oct 14 2017 *)

a(n) = sumdiv(n, d, moebius(d)*11^(n/d))/n; \\ Michel Marcus, Jan 07 2015

"CHK" (necklace, identity, unlabeled) transform of 11, 0, 0, 0...
a(n) = Sum_{d|n} mu(d)*11^(n/d)/n.
G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 11*x^k))/k. - Ilya Gutkovskiy, May 19 2019

A032167 Number of aperiodic necklaces of n beads of 12 colors.

Original entry on oeis.org

12, 66, 572, 5148, 49764, 497354, 5118828, 53745120, 573308736, 6191711526, 67546215516, 743008120140, 8230246567620, 91708459194066, 1027134771622388, 11555266154065920, 130506535690613940

Offset: 1

Christian G. Bower

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..500
C. G. Bower, Transforms (2)
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
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]
Index entries for sequences related to Lyndon words

Column 12 of A074650.

f[d_]:=MoebiusMu[d] 12^(n/d)/n; a[n_]:=Total[f/@Divisors[n]]; a[0]=1; Table[a[n], {n, 1, 30}] (* Vincenzo Librandi, Oct 14 2017 *)

"CHK" (necklace, identity, unlabeled) transform of 12, 0, 0, 0...
a(n) = Sum_{d|n} mu(d)*12^(n/d)/n.
G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 12*x^k))/k. - Ilya Gutkovskiy, May 19 2019

cp's OEIS Frontend

A032242 Number of identity bracelets of n beads of 5 colors.

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A136704 Number of Lyndon words on {1,2,3} with an odd number of 1's and an odd number of 2's.

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A261531 Number of necklaces with n beads of unlabeled colors such that the numbers of beads per color are distinct.

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A261599 Number of primitive (aperiodic, or Lyndon) necklaces with n beads of unlabeled colors such that the numbers of beads per color are distinct.

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

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A005516 Number of n-bead bracelets (turnover necklaces) with 12 red beads.

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Mathematica

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A005654 Number of bracelets (turn over necklaces) with n red, 1 pink and n-1 blue beads; also reversible strings with n red and n-1 blue beads; also next-to-central column in Losanitsch's triangle A034851.

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