A032192 -id:A032192 - 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

A053618 a(n) = ceiling(binomial(n,4)/n).

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 5, 9, 14, 21, 30, 42, 55, 72, 91, 114, 140, 170, 204, 243, 285, 333, 385, 443, 506, 575, 650, 732, 819, 914, 1015, 1124, 1240, 1364, 1496, 1637, 1785, 1943, 2109, 2285, 2470, 2665, 2870, 3086, 3311, 3548, 3795, 4054, 4324

Offset: 1

N. J. A. Sloane, Mar 25 2000

Colin Barker, Table of n, a(n) for n = 1..1000
R. L. Graham and N. J. A. Sloane, Lower bounds for constant weight codes, IEEE Trans. Inform. Theory, 26 (1980), 37-43.
Index entries for linear recurrences with constant coefficients, signature (3,-3,1,0,0,0,0,1,-3,3,-1).

Cf. A007997, A008646, A032192, A053643, A053731, A053733.

[Ceiling(Binomial(n,4)/n): n in [1..60]]; // G. C. Greubel, May 16 2019

CoefficientList[Series[x^4*(1-x+x^2)*(1-x+x^2+x^4)/((1-x)^3*(1-x^8)), {x,0,60}], x] (* G. C. Greubel, May 16 2019 *)

concat([0,0,0], Vec(x^4*(x^2-x+1)*(x^4+x^2-x+1) / ((x-1)^4*(x+1)*(x^2+1)*(x^4+1)) + O(x^60))) \\ Colin Barker, Jan 20 2015

[ceil(binomial(n,4)/n) for n in (1..60)] # G. C. Greubel, May 16 2019

a(n) = ( 2*n^3 - 12*n^2 + 22*n - 3 + 9*(-1)^n + 3*(1+(-1)^n)*(-1)^(n*(n-1)/2) - 6*(1 + (-1)^n)*(-1)^floor(n/4) )/48. - Luce ETIENNE, Jan 20 2015

G.f.: x^4*(1 - x + x^2)*(1 - x + x^2 + x^4)/((1-x)^3*(1-x^8)). - Colin Barker, Jan 20 2015

A032193 Number of necklaces with 8 black beads and n-8 white beads.

Original entry on oeis.org

1, 1, 5, 15, 43, 99, 217, 429, 810, 1430, 2438, 3978, 6310, 9690, 14550, 21318, 30667, 43263, 60115, 82225, 111041, 148005, 195143, 254475, 328756, 420732, 534076, 672452, 840652, 1043460, 1287036, 1577532, 1922741

Offset: 8

Christian G. Bower

nonn

The g.f. is Z(C_8,x)/x^8, the 8-variate cycle index polynomial for the cyclic group C_8, with substitution x[i]->1/(1-x^i), i=1,...,8. Therefore by Polya enumeration a(n+8) is the number of cyclically inequivalent 8-necklaces whose 8 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_8,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - Wolfdieter Lang, Feb 15 2005
From Petros Hadjicostas, Aug 31 2018: (Start)
The CIK[k] transform of sequence (c(n): n>=1) has generating function A_k(x) = (1/k)*Sum_{d|k} phi(d)*C(x^d)^{k/d}, where C(x) = Sum_{n>=1} c(n)*x^n is the g.f. of (c(n): n>=1).
When c(n) = 1 for all n >= 1, we get C(x) = x/(1-x) and A_k(x) = (x^k/k)*Sum_{d|k} phi(d)*(1-x^d)^{-k/d}, which is the g.f. of the number a_k(n) of necklaces of n beads of 2 colors with k of them black and n-k of them white.
Using Taylor expansions, we can easily prove that a_k(n) = (1/k)*Sum_{d|gcd(n,k)} phi(d)*binomial(n/d - 1, k/d - 1) = (1/n)*Sum_{d|gcd(n,k)} phi(d)*binomial(n/d, k/d), which is Robert A. Russell's formula in the Mathematica code below.
For this sequence k = 8, and thus we get the formulae below.
(End)

Christian G. Bower, Transforms (2)
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
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]
Index entries for sequences related to necklaces
Index entries for linear recurrences with constant coefficients, signature (4,-4,-4,11,-8,0,8,-10,0,8,0,-10,8,0,-8,11,-4,-4,4,-1).

Column k=8 of A047996.

Cf. A004526, A005514, A007997, A008610, A008646, A032191, A032192.

k = 8; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)
CoefficientList[Series[1/8*(1/(1 - x)^8 + 1/(1 - x^2)^4 + 2/(1 - x^4)^2 + 4/(1 - x^8)^1),{x, 0, 30}], x] (* Stefano Spezia, Sep 01 2018 *)

"CIK[ 8 ]" (necklace, indistinct, unlabeled, 8 parts) transform of 1, 1, 1, 1...
G.f.: (x^8)*(1-3*x+5*x^2+3*x^3-4*x^4+4*x^5+6*x^6-4*x^7+7*x^8-x^9+x^10+x^11)/((1-x)^4*(1-x^2)^2*(1-x^4)*(1-x^8)).
G.f.: 1/8*x^8*(1/(1-x)^8+1/(1-x^2)^4+2/(1-x^4)^2+4/(1-x^8)^1). - Herbert Kociemba, Oct 22 2016
a(n) = (1/8)*Sum_{d|gcd(n,8)} phi(d)*binomial(n/d - 1, 8/d - 1) = (1/n)*Sum_{d|gcd(n,8)} phi(d)*binomial(n/d, 8/d). - Petros Hadjicostas, Aug 31 2018

A053731 a(n) = ceiling(binomial(n,8)/n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 15, 42, 99, 215, 429, 805, 1430, 2431, 3978, 6299, 9690, 14535, 21318, 30645, 43263, 60088, 82225, 111004, 148005, 195098, 254475, 328697, 420732, 534006, 672452, 840565, 1043460, 1286934, 1577532, 1922618

Offset: 1

N. J. A. Sloane, Mar 25 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
R. L. Graham and N. J. A. Sloane, Lower bounds for constant weight codes, IEEE Trans. Inform. Theory, 26 (1980), 37-43.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-7,21,-35,35,-21,7,-1).

Cf. Sequences of the form ceiling(binomial(n,k)/n): A000012 (k=1), A004526 (k=2), A007997 (k=3), A008646 (k=5), A032192 (k=7), A053618 (k=4), A053643 (k=6), this sequence (k=8), A053733 (k=9).

[Ceiling(Binomial(n,8)/n): n in [1..45]]; // G. C. Greubel, Sep 06 2019

seq(ceil(binomial(n,8)/n), n=1..45); # G. C. Greubel, Sep 06 2019

Table[Ceiling[Binomial[n, 8]/n], {n, 45}] (* G. C. Greubel, Sep 06 2019 *)

vector(45, n, ceil(binomial(n,8)/n)) \\ G. C. Greubel, Sep 06 2019

[ceil(binomial(n,8)/n) for n in (1..45)] # G. C. Greubel, Sep 06 2019

A032194 Number of necklaces with 9 black beads and n-9 white beads.

Original entry on oeis.org

1, 1, 5, 19, 55, 143, 335, 715, 1430, 2704, 4862, 8398, 14000, 22610, 35530, 54484, 81719, 120175, 173593, 246675, 345345, 476913, 650325, 876525, 1168710, 1542684, 2017356, 2615104, 3362260, 4289780, 5433736, 6835972

Offset: 9

Christian G. Bower

nonn

The g.f. is Z(C_9,x)/x^9, the 9-variate cycle index polynomial for the cyclic group C_9, with substitution x[i]->1/(1-x^i), i=1,...,9. Therefore by Polya enumeration a(n+9) is the number of cyclically inequivalent 9-necklaces whose 9 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_9,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - Wolfdieter Lang, Feb 15 2005

Christian G. Bower, Transforms (2)
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
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]
Index entries for sequences related to necklaces
Index entries for linear recurrences with constant coefficients, signature (6,-15,22,-27,36,-42,36,-27,23,-21,21,-23,27,-36,42,-36,27,-22,15,-6,1).

Column k=9 of A047996.

Cf. A004526, A007997, A008610, A008646, A032191, A032192, A032193.

k = 9; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)

"CIK[ 9 ]" (necklace, indistinct, unlabeled, 9 parts) transform of 1, 1, 1, 1...
G.f.: (x^9)*(1-5*x+14*x^2-18*x^3+21*x^4-21*x^5+25*x^6 -21*x^7 +21*x^8 -18*x^9 +14*x^10 -5*x^11 +x^12) / ((1-x)^6*(1-x^3)^2*(1-x^9)).
G.f.: (1/9)*x^9*(1/(1-x)^9+2/(1-x^3)^3+6/(1-x^9)^1). - Herbert Kociemba, Oct 22 2016

A053733 a(n) = ceiling(binomial(n,9)/n).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 5, 19, 55, 143, 334, 715, 1430, 2702, 4862, 8398, 13997, 22610, 35530, 54480, 81719, 120175, 173587, 246675, 345345, 476905, 650325, 876525, 1168700, 1542684, 2017356, 2615092, 3362260, 4289780, 5433722

Offset: 1

N. J. A. Sloane, Mar 25 2000

nonn

G. C. Greubel, Table of n, a(n) for n = 1..1000
R. L. Graham and N. J. A. Sloane, Lower bounds for constant weight codes, IEEE Trans. Inform. Theory, 26 (1980), 37-43.
Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-8,28,-56,70,-56,28,-8,1).

Cf. Sequences of the form ceiling(binomial(n,k)/n): A000012 (k=1), A004526 (k=2), A007997 (k=3), A008646 (k=5), A032192 (k=7), A053618 (k=4), A053643 (k=6), A053731 (k=8), this sequence (k=9).

[Ceiling(Binomial(n,9)/n): n in [1..40]]; // G. C. Greubel, Sep 06 2019

seq(ceil(binomial(n,9)/n), n=1..40); # G. C. Greubel, Sep 06 2019

Table[Ceiling[Binomial[n, 9]/n], {n, 40}] (* G. C. Greubel, Sep 06 2019 *)

vector(40, n, ceil(binomial(n,9)/n)) \\ G. C. Greubel, Sep 06 2019

[ceil(binomial(n,9)/n) for n in (1..40)] # G. C. Greubel, Sep 06 2019

A347974 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_8)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 17, 17, 1, 1, 47, 242, 47, 1, 1, 113, 3071, 3071, 113, 1, 1, 245, 34477, 232290, 34477, 245, 1, 1, 491, 341633, 16665755, 16665755, 341633, 491, 1, 1, 920, 3022045, 1073874283, 8241549097, 1073874283, 3022045, 920, 1, 1, 1635, 24145695

Offset: 0

Álvar Ibeas, Sep 21 2021

Columns can be computed by a method analogous to that of Fripertinger for isometry classes of linear codes, disallowing scalar transformation of individual coordinates.

Regarding the formula for column k = 1, note that A241926(q - 1, n) counts, up to coordinate permutation, one-dimensional subspaces of (F_q)^n generated by a vector with no zero component.

			Triangle begins:
  k:  0    1    2    3    4    5
      --------------------------
n=0:  1
n=1:  1    1
n=2:  1    5    1
n=3:  1   17   17    1
n=4:  1   47  242   47    1
n=5:  1  113 3071 3071  113    1
There are 9 = A022172(2, 1) one-dimensional subspaces in (F_8)^2. Among them, <(1, 1)> is invariant by coordinate swap and the rest are grouped in orbits of size two. Hence, T(2, 1) = 5.

Álvar Ibeas, Entries up to T(10, 4)
H. Fripertinger, Isometry classes of codes
Álvar Ibeas, Column k=1 up to n=100
Álvar Ibeas, Column k=2 up to n=100
Álvar Ibeas, Column k=3 up to n=100
Álvar Ibeas, Column k=4 up to n=100

Cf. A022172, A032192, A241926.

T(n, 1) = T(n - 1, 1) + A032192(n + 7).

A032195 Number of necklaces with 10 black beads and n-10 white beads.

Original entry on oeis.org

1, 1, 6, 22, 73, 201, 504, 1144, 2438, 4862, 9252, 16796, 29414, 49742, 81752, 130752, 204347, 312455, 468754, 690690, 1001603, 1430715, 2016144, 2804880, 3856892, 5245128, 7060984, 9414328, 12440668, 16301164

Offset: 10

Christian G. Bower

nonn

The g.f. is Z(C_10,x)/x^10, the 10-variate cycle index polynomial for the cyclic group C_10, with substitution x[i]->1/(1-x^i), i=1,...,10. By Polya enumeration, a(n+10) is the number of cyclically inequivalent 10-necklaces whose 10 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_10,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - Wolfdieter Lang, Feb 15 2005

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 necklaces
Index entries for linear recurrences with constant coefficients, signature (4,-2,-12,17,9,-32,10,29,-29,-9,28,-7,-5,-5,-7,28,-9,-29,29,10,-32,9,17,-12,-2,4,-1).

Column k=10 of A047996.

Cf. A004526, A007997, A008610, A008646, A032191, A032192, A032193, A032194.

k = 10; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)

"CIK[ 10 ]" (necklace, indistinct, unlabeled, 10 parts) transform of 1, 1, 1, 1...
G.f.: (x^10)*(1-3*x+4*x^2+12*x^3-8*x^4-x^5+31*x^6-4*x^8+16*x^9 +11*x^10 +3*x^11+8*x^12+4*x^13+4*x^14+x^15+x^16) /((1-x)^4*(1-x^2)^4 *(1-x^5)*(1-x^10)).
G.f.: (1/10)*x^10*(1/(1 - x)^10 + 1/(1 - x^2)^5 + 4/(1 - x^5)^2 + 4/(1 - x^10)^1). - Herbert Kociemba, Oct 22 2016

A032196 Number of necklaces with 11 black beads and n-11 white beads.

Original entry on oeis.org

1, 1, 6, 26, 91, 273, 728, 1768, 3978, 8398, 16796, 32066, 58786, 104006, 178296, 297160, 482885, 766935, 1193010, 1820910, 2731365, 4032015, 5864750, 8414640, 11920740, 16689036, 23107896, 31666376, 42975796

Offset: 11

Christian G. Bower

nonn

The g.f. is Z(C_11,x)/x^11, the 11-variate cycle index polynomial for the cyclic group C_11, with substitution x[i]->1/(1-x^i), i=1..11. By Polya enumeration, a(n+11) is the number of cyclically inequivalent 11-necklaces whose 11 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_11,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - Wolfdieter Lang, Feb 15 2005

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 necklaces
Index entries for linear recurrences with constant coefficients, signature (10,-45,120,-210,252,-210,120,-45,10,-1,1,-10,45,-120,210,-252,210,-120,45,-10,1).

Column k=11 of A047996.

Cf. A004526, A007997, A008610, A008646, A032191, A032192, A032193, A032194, A032195.

k = 11; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)
DeleteCases[CoefficientList[Series[(x^11) (1 - 9 x + 41 x^2 - 109 x^3 + 191 x^4 - 229 x^5 + 191 x^6 - 109 x^7 + 41 x^8 - 9 x^9 + x^10)/((1 - x)^10 (1 - x^11)), {x, 0, 39}], x], 0] (* Michael De Vlieger, Oct 10 2016 *)

"CIK[ 11 ]" (necklace, indistinct, unlabeled, 11 parts) transform of 1, 1, 1, 1...
G.f.: (x^11) * (1 - 9*x + 41*x^2 - 109*x^3 + 191*x^4 - 229*x^5 + 191*x^6 - 109*x^7 + 41*x^8 - 9*x^9 + x^10) / ((1-x)^10 * (1-x^11)).
a(n) = ceiling(binomial(n, 11)/n) (conjecture Wolfdieter Lang).
From Herbert Kociemba, Oct 11 2016: (Start)
This conjecture indeed is true.
Sketch of proof:
There are binomial(n,11) ways to place the 11 black beads in the necklace with n beads. If n is not divisible by 11 there are no necklaces with a rotational symmetry. So exactly n necklaces are equivalent up to rotation and there are binomial(n,11)/n = ceiling(binomial(n,11)/n) equivalence classes.
If n is divisible by 11 the only way to get a necklace with rotational symmetry is to space out the 11 black beads evenly. There are n/11 ways to do this and all ways are equivalent up to rotation. So there are  binomial(n,11) - n/11 unsymmetric necklaces which give binomial(n,11)/n - 1/11 equivalence classes. If we add the single symmetric equivalence class we get Binomial(n,11)/n - 1/11 + 1 which also is ceiling(binomial(n,11)/n). (End)
G.f.: (10/(1 - x^11) + 1/(1 - x)^11)*x^11/11. - Herbert Kociemba, Oct 16 2016

A032197 Number of necklaces with 12 black beads and n-12 white beads.

Original entry on oeis.org

1, 1, 7, 31, 116, 364, 1038, 2652, 6310, 14000, 29414, 58786, 112720, 208012, 371516, 643856, 1086601, 1789515, 2883289, 4552275, 7056280, 10752060, 16128424, 23841480, 34769374, 50067108, 71250060, 100276894, 139672312

Offset: 12

Christian G. Bower

nonn

The g.f. is Z(C_12,x)/x^12, the 12-variate cycle index polynomial for the cyclic group C_12, with substitution x[i]->1/(1-x^i), i=1,...,12. Therefore by Polya enumeration a(n+12) is the number of cyclically inequivalent 12-necklaces whose 12 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_12,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - Wolfdieter Lang, Feb 15 2005

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 necklaces
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 A047996.

Cf. A004526, A007997, A008610, A008646, A032191, A032192-A032196.

k = 12; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)

"CIK[ 12 ]" (necklace, indistinct, unlabeled, 12 parts) transform of 1, 1, 1, 1...
G.f.: (x^12)*(1-3*x+7*x^2+9*x^3+18*x^4+38*x^5+72*x^6+92*x^7+168*x^8+160*x^9+238*x^10+230*x^11+296*x^12+234*x^13+330*x^14+248*x^15+284*x^16+238*x^17+230*x^18+166*x^19+172*x^20+78*x^21+80*x^22+38*x^23+21*x^24+7*x^25+3*x^26+x^27) /((1+x)*(1-x)*(1-x^2)*(1-x^3)*(1-x)^5*(1+x+x^2)*(1-x^4)^2*(1-x^6)*(1-x^12)). - Wolfdieter Lang, Feb 15 2005 (see comment)
G.f.: 1/12 x^12 ((1 - x)^-12 + (1 - x^2)^-6 + 2 (1 - x^3)^-4 + 2 (1 - x^4)^-3 + 2 (1 - x^6)^-2 + 4 (1 - x^12)^-1). - Herbert Kociemba, Oct 22 2016

cp's OEIS Frontend

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

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A053618 a(n) = ceiling(binomial(n,4)/n).

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A032193 Number of necklaces with 8 black beads and n-8 white beads.

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A053731 a(n) = ceiling(binomial(n,8)/n).

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A032194 Number of necklaces with 9 black beads and n-9 white beads.

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A053733 a(n) = ceiling(binomial(n,9)/n).

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A347974 Triangle read by rows: T(n, k) is the number of k-dimensional subspaces in (F_8)^n, counted up to coordinate permutation (n >= 0, 0 <= k <= n).

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