A032281 -id:A032281 - cp's OEIS frontend

A052307 Triangle read by rows: T(n,k) = number of bracelets (reversible necklaces) with n beads, k of which are black and n - k are white.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 3, 3, 3, 1, 1, 1, 1, 3, 4, 4, 3, 1, 1, 1, 1, 4, 5, 8, 5, 4, 1, 1, 1, 1, 4, 7, 10, 10, 7, 4, 1, 1, 1, 1, 5, 8, 16, 16, 16, 8, 5, 1, 1, 1, 1, 5, 10, 20, 26, 26, 20, 10, 5, 1, 1, 1, 1, 6, 12, 29, 38, 50, 38, 29, 12, 6, 1, 1, 1, 1, 6, 14, 35, 57, 76, 76, 57, 35, 14, 6, 1, 1

Offset: 0

Christian G. Bower, Nov 15 1999

Equivalently, T(n,k) is the number of orbits of k-element subsets of the vertices of a regular n-gon under the usual action of the dihedral group D_n, or under the action of Euclidean plane isometries. Note that each row of the table is symmetric and unimodal. - Austin Shapiro, Apr 20 2009

Also, the number of k-chords in n-tone equal temperament, up to (musical) transposition and inversion. Example: there are 29 tetrachords, 38 pentachords, 50 hexachords in the familiar 12-tone equal temperament. Called "Forte set-classes," after Allen Forte who first cataloged them. - Jon Wild, May 21 2004

			Triangle T(n,k) (with rows n >= 0 and columns k = 0..n) begins:
   1;
   1,  1;
   1,  1,  1;
   1,  1,  1,  1;
   1,  1,  2,  1,  1;
   1,  1,  2,  2,  1,  1;
   1,  1,  3,  3,  3,  1,  1;
   1,  1,  3,  4,  4,  3,  1,  1;
   1,  1,  4,  5,  8,  5,  4,  1,  1;
   1,  1,  4,  7, 10, 10,  7,  4,  1,  1;
   1,  1,  5,  8, 16, 16, 16,  8,  5,  1,  1;
   1,  1,  5, 10, 20, 26, 26, 20, 10,  5,  1,  1;
   1,  1,  6, 12, 29, 38, 50, 38, 29, 12,  6,  1,  1;
   ...

Martin Gardner, "New Mathematical Diversions from Scientific American" (Simon and Schuster, New York, 1966), pages 245-246.
N. Zagaglia Salvi, Ordered partitions and colourings of cycles and necklaces, Bull. Inst. Combin. Appl., 27 (1999), 37-40.

Washington Bomfim, Rows n = 0..130, flattened
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.
G. Gori, S. Paganelli, A. Sharma, P. Sodano, and A. Trombettoni, Bell-Paired States Inducing Volume Law for Entanglement Entropy in Fermionic Lattices, arXiv:1405.3616 [cond-mat.stat-mech], 2014. See Section V.
H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10(8) (1979), 964-999.
S. Karim, J. Sawada, Z. Alamgirz, and S. M. Husnine, Generating bracelets with fixed content, Theoretical Computer Science, 475 (2013), 103-112.
John P. McSorley and Alan H. Schoen, Rhombic tilings of (n, k)-ovals, (n, k, lambda)-cyclic difference sets, and related topics, Discrete Math., 313 (2013), 129-154. - From _N. J. A. Sloane_, Nov 26 2012
A. L. Patterson, Ambiguities in the X-Ray Analysis of Crystal Structures, Phys. Rev. 65 (1944), 195 - 201 (see Table I). [From _N. J. A. Sloane_, Mar 14 2009]
Richard H. Reis, A formula for C(T) in Gupta's paper, Indian J. Pure and Appl. Math., 10(8) (1979), 1000-1001.
Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35(5) (2004), 629-638.
Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., 35(5) (2004), 629-638.
Index entries for sequences related to bracelets

Row sums: A000029. Columns 0-12: A000012, A000012, A008619, A001399, A005232, A032279, A005513, A032280, A005514, A032281, A005515, A032282, A005516.

Cf. A047996, A051168, A052308, A052309, A052310.

A052307 := proc(n,k)
        local hk,a,d;
        if k = 0 then
                return 1 ;
        end if;
        hk := k mod 2 ;
        a := 0 ;
        for d in numtheory[divisors](igcd(k,n)) do
                a := a+ numtheory[phi](d)*binomial(n/d-1,k/d-1) ;
        end do:
        %/k + binomial(floor((n-hk)/2),floor(k/2)) ;
        %/2 ;
end proc: # R. J. Mathar, Sep 04 2011

Table[If[m*n===0,1,1/2*If[EvenQ[n], If[EvenQ[m], Binomial[n/2, m/2], Binomial[(n-2)/2, (m-1)/2 ]], If[EvenQ[m], Binomial[(n-1)/2, m/2], Binomial[(n-1)/2, (m-1)/2]]] + 1/2*Fold[ #1 +(EulerPhi[ #2]*Binomial[n/#2, m/#2])/n &, 0, Intersection[Divisors[n], Divisors[m]]]], {n,0,12}, {m,0,n}] (* Wouter Meeussen, Aug 05 2002, Jan 19 2009 *)

B(n,k)={ if(n==0, return(1)); GCD = gcd(n, k); S = 0;
for(d = 1, GCD, if((k%d==0)&&(n%d==0), S+=eulerphi(d)*binomial(n/d,k/d)));
return (binomial(floor(n/2)- k%2*(1-n%2), floor(k/2))/2 + S/(2*n)); }
n=0;k=0; for(L=0, 8645, print(L, " ", B(n,k)); k++; if(k>n, k=0; n++))
/* Washington Bomfim, Jun 30 2012 */

from sympy import binomial as C, totient, divisors, gcd
def T(n, k): return 1 if n==0 else C((n//2) - k%2 * (1 - n%2), (k//2))/2 + sum(totient(d)*C(n//d, k//d) for d in divisors(gcd(n, k)))/(2*n)
for n in range(11): print([T(n, k) for k in range(n + 1)]) # Indranil Ghosh, Apr 23 2017

T(0,0) = 1. If n > 0, T(n,k) = binomial(floor(n/2) - (k mod 2) * (1 - (n mod 2)), floor(k/2)) / 2 + Sum_{d|n, d|k} (phi(d)*binomial(n/d, k/d)) / (2*n). - Washington Bomfim, Jun 30 2012 [edited by Petros Hadjicostas, May 29 2019]
From Freddy Barrera, Apr 21 2019: (Start)
T(n,k) = (1/2) * (A119963(n,k) + A047996(n,k)).
T(n,k) = T(n, n-k) for each k < n (Theorem 2 of H. Gupta). (End)
G.f. for column k >= 1: (x^k/2) * ((1/k) * Sum_{m|k} phi(m)/(1 - x^m)^(k/m) + (1 + x)/(1 - x^2)^floor((k/2) + 1)). (This formula is due to Herbert Kociemba.) - Petros Hadjicostas, May 25 2019
Bivariate o.g.f.: Sum_{n,k >= 0} T(n, k)*x^n*y^k = (1/2) * ((x + 1) * (x*y + 1) / (1 - x^2 * (y^2 + 1)) + 1 - Sum_{d >= 1} (phi(d)/d) * log(1 - x^d * (1 + y^d))). - Petros Hadjicostas, Jun 13 2019

A032279 Number of bracelets (turnover necklaces) of n beads of 2 colors, 5 of them black.

Original entry on oeis.org

1, 1, 3, 5, 10, 16, 26, 38, 57, 79, 111, 147, 196, 252, 324, 406, 507, 621, 759, 913, 1096, 1298, 1534, 1794, 2093, 2421, 2793, 3199, 3656, 4152, 4706, 5304, 5967, 6681, 7467, 8311, 9234, 10222, 11298, 12446, 13691, 15015, 16445

Offset: 5

Christian G. Bower, N. J. A. Sloane

From Vladimir Shevelev, Apr 23 2011: (Start)
Also number of non-equivalent necklaces of 5 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=5. The full solution was given by H. Gupta (1979); I gave a short proof of Gupta's result and showed an equivalence of this problem and every one of the following problems: enumerating the bracelets of n beads of 2 colors, k of them black, and enumerating the necklaces of k beads each of them 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 n with five 1's in every row. (End)
a(n+5) is the number of symmetry-allowed, linearly-independent terms at n-th order in the series expansion of the T_1 X h vibronic perturbation matrix, H(Q) (cf. Dunn & Bates). - Bradley Klee, Jul 20 2015
From Petros Hadjicostas, Jul 17 2018: (Start)
Let (c(n): n >= 1) be a sequence of nonnegative integers and let C(x) = Sum_{n>=1} c(n)*x^n be its g.f. Let k be a positive integer. Let a_k = (a_k(n): n >= 1) be the output sequence of the DIK[k] transform of sequence (c(n): n >= 1), and let A_k(x) = Sum_{n>=1} a_k(n)*x^n be its g.f. See Christian G. Bower's web link below. It can be proved that, when k is odd, A_k(x) = ((1/k)*Sum_{d|k} phi(d)*C(x^d)^(k/d) + C(x^2)^((k-1)/2)*C(x))/2.
For this sequence, k = 5, c(n) = 1 for all n >= 1, and C(x) = x/(1-x). Thus, a(n) = a_5(n) for all n >= 1. Since a_k(n) = 0  for 1 <= n <= k-1, the offset of this sequence is n = k = 5. Applying the formula for the g.f. of DIK[5] of (c(n): n >= 1) with C(x) = x/(1-x) and k = 5, we get A(x) = A_5(x) = x^5*((1/5)*Sum_{d|5} phi(d)*(1-x^d)^(-5/d) + (1+x)/(1-x^2)^3)/2, which obviously equals the g.f. in the formula section below.
The g.f. is also a special case of Herbert Kociemba's formula that is valid for both even and odd k: A_k(x) = x^k*((1/k)*Sum_{d|k} phi(d)*(1-x^d)^(-k/d) + (1+x)/(1-x^2)^Floor[(k+2)/2])/2.
Here, a(n) is defined to be the number of n-bead bracelets of two colors with 5 black beads and n-5 white beads. But it is also the number of dihedral compositions of n with 5 positive parts. (This statement is equivalent to Vladimir Shevelev's statement above that a(n) is the "number of non-equivalent necklaces of 5 beads each of them painted by one of n colors." By "necklaces" he means "turnover necklaces". See paragraph (2) of Section 2 in his 2004 paper in the Indian Journal of Pure and Applied Mathematics.)
Two cyclic compositions of n (with k = 5 parts) belong to the same equivalence class corresponding to a dihedral composition of n if and only if one can be obtained from the other by a rotation or reversal of order. (End)

			From _Petros Hadjicostas_, Jul 17 2018: (Start)
Every n-bead bracelet of two colors such that 5 beads are black and n-5 are white can be transformed into a dihedral composition of n with 5 positive parts in the following way. Start with one B bead and go in one direction (say clockwise) until you reach the next B bead. Continue this process until you come back to the original B bead.
Let b_i be the number of beads from B bead i until you reach the last W bead before B bead i+1 (or B bead 1). Here, b_i = 1 iff there are no W beads between B bead i and B bead i+1 (or B bead 5 and B bead 1). Then b_1 + b_2 + b_3 + b_4 + b_5 = n, and we get a dihedral composition of n. (Of course, b_2 + b_3 + b_4 + b_5 + b_1 and b_5 + b_4 + b_3 + b_2 + b_1 belong to the same equivalence class of the dihedral composition b_1 + b_2 + b_3 + b_4 + b_5.)
For example, a(8) = 5, and we have the following bracelets with 5 B beads and 3 W beads. Next to the bracelets we list the corresponding dihedral compositions of n with k=5 parts (they must be viewed on a circle):
BBBBBWWW <-> 1+1+1+1+4
BBBBWBWW <-> 1+1+1+2+3
BBWBBBWW <-> 1+2+1+1+3
BWBBWBWB <-> 2+1+2+2+1
BWBWBWBB <-> 2+2+2+1+1
(End)

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 = 5..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.
N. Benyahia Tani, Z. Yahi, and S. Bouroubi, Ordered and non-ordered non-isometric convex quadrilaterals inscribed in a regular n-gon, Bulletin du Laboratoire Liforce, 01 (2014) 1-9.
C. G. Bower, Transforms (2)
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.
J. L. Dunn and C. A. Bates, Analysis of the T1u(x)hg system as a model for C60 molecules, Phys. Rev. B 52, 5996, 15 August 1995.
H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., Vol. 10, No. 8 (1979), 964-999.
E. Kirkman, J. Kuzmanovich, and J. J. Zhang, Invariants of (-1)-Skew Polynomial Rings under Permutation Representations, arXiv preprint arXiv:1305.3973, 2013. See Example 5.5.
Richard H. Reis, A formula for C(T) in Gupta's paper, Indian J. Pure and Appl. Math., Vol. 10 , No. 8 (1979), 1000-1001.
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., Vol. 35, No. 5 (2004), 629-638.
Vladimir Shevelev, Necklaces and convex k-gons, Indian J. Pure and Appl. Math., Vol. 35, No. 5 (2004), 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), arXiv:1104.4051 [math.CO], 2011. (Cf. Section 5).
Index entries for sequences related to bracelets
Index entries for linear recurrences with constant coefficients, signature (2,1,-4,1,3,-3,-1,4,-1,-2,1).

Column k=5 of A052307.

Cf. A005514, A008805, A032280, A032281, A032282, A292906.

m:=50; R:=PowerSeriesRing(Integers(), m); Coefficients(R!(1-x+2*x^3-x^5+x^6)/((1-x)^2*(1-x^2)^2*(1-x^5))); // Vincenzo Librandi, Sep 07 2013

seq(floor(n^4/240 + n^3/24 + 5*n^2/24 + 25*n/48 + 1 + (-1)^n*n/16), n=0..100); # Robert Israel, Jul 22 2015

k = 5; 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 + 2 x^3 - x^5 + x^6) / ((1 - x)^2 (1 - x^2)^2 (1 - x^5)), {x, 0, 50}], x] (* Vincenzo Librandi, Sep 07 2013 *)
k=5 (* Number of black beads in bracelet problem *); 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 *)

a(n) = round((n^4 -10*n^3 +50*n^2 -(110+30*(1-n%2))*n)/240 +3/5) \\ Washington Bomfim, Jul 17 2008

"DIK[ 5 ]" (necklace, indistinct, unlabeled, 5 parts) transform of 1, 1, 1, 1, ...
G.f.: x^5*(1-x+2*x^3-x^5+x^6)/((1-x)^2*(1-x^2)^2*(1-x^5)). - corrected for offset 5 by Robert Israel, Jul 22 2015
From Vladimir Shevelev, Apr 23 2011: (Start)
Put s(n,k,d)=1, if n == k (mod d), and 0, otherwise. Then
a(n) = (2/5)*s(n,0,5) + (n-1)*(n-3)*((n-2)*(n-4) + 15)/240, if n is odd >= 5;
a(n) = (2/5)*s(n,0,5) + (n-2)*(n-4)*((n-1)*(n-3) + 15)/240, if n is even >= 5. (End)
a(n+5) = floor(n^4/240 + n^3/24 + 5*n^2/24 + 25*n/48 + 1 + (-1)^n*n/16). - Robert Israel, Jul 22 2015
a(n) = (A008646(n-5) + A119963(n, 5))/2 = (A008646(n-5) + C(floor((n-1)/2), 2))/2 for n >= 5. - Petros Hadjicostas, Jul 17 2018

A189976 a(n) is the number of incongruent two-color bracelets of n beads, 8 of them black (A005514), having a diameter of symmetry.

Original entry on oeis.org

1, 1, 5, 5, 15, 15, 35, 35, 70, 70, 126, 126, 210, 210, 330, 330, 495, 495, 715, 715, 1001, 1001, 1365, 1365, 1820, 1820, 2380, 2380, 3060, 3060, 3876, 3876, 4845, 4845, 5985, 5985, 7315, 7315, 8855, 8855, 10626

Offset: 8

Vladimir Shevelev, May 03 2011

For n >= 9, a(n-1) is the number of incongruent two-color bracelets of n beads, 9 from them are black (A032281), having a diameter of symmetry.

Vincenzo Librandi, Table of n, a(n) for n = 8..1000
H. Gupta, Enumeration of incongruent cyclic k-gons, Indian J. Pure and Appl. Math., 10 (1979), no.8, 964-999.
V. Shevelev, A problem of enumeration of two-color bracelets with several variations, arXiv:0710.1370 [math.CO], 2007-2011.
Index entries for linear recurrences with constant coefficients, signature (1,4,-4,-6,6,4,-4,-1,1).

Cf. A005514, A032281, A008805, A058187.

[Binomial(Floor(n/2),4): n in[8..60]]; // Vincenzo Librandi, Aug 10 2014

A189976 :=proc(n): binomial(floor(n/2),4) end: seq(A189976(n), n=8..48); # Johannes W. Meijer, Aug 15 2011

Module[{c=Binomial[Range[4,30],4]},Riffle[c,c]] (* Harvey P. Dale, Aug 09 2014 *)
Table[(Binomial[Floor[n/2], 4]), {n, 8, 40}] (* Vincenzo Librandi, Aug 10 2014 *)

a(n) = C(floor(n/2),4).
a(n+5) = A194005(n,n-4). [Johannes W. Meijer, Aug 15 2011]
G.f.: -x^8/((x-1)^5*(x+1)^4). [Colin Barker, Feb 06 2013]

Data added and link corrected by Johannes W. Meijer, Aug 15 2011

A005514 Number of n-bead bracelets (turnover necklaces) with 8 red beads and n-8 black beads.

Original entry on oeis.org

1, 1, 5, 10, 29, 57, 126, 232, 440, 750, 1282, 2052, 3260, 4950, 7440, 10824, 15581, 21879, 30415, 41470, 56021, 74503, 98254, 127920, 165288, 211276, 268228, 337416, 421856, 523260, 645456, 790704, 963793, 1167645, 1408185

Offset: 8

N. J. A. Sloane

From Vladimir Shevelev, Apr 23 2011: (Start)
Also number of non-equivalent necklaces of 8 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=8 (see our comment at A032279). (End)
From Petros Hadjicostas, Jul 14 2018: (Start)
Let (c(n): n >= 1) be a sequence of nonnegative integers and let C(x) = Sum_{n>=1} c(n)*x^n be its g.f. Let k be a positive integer. Let a_k = (a_k(n): n >= 1) be the output sequence of the DIK[k] transform of sequence (c(n): n >= 1), and let A_k(x) = Sum_{n>=1} a_k(n)*x^n be its g.f. See Bower's web link below. It can be proved that, when k is even, A_k(x) = ((1/k)*Sum_{d|k} phi(d)*C(x^d)^(k/d) + (1/2)*C(x^2)^((k/2)-1)*(C(x)^2 + C(x^2)))/2.
For this sequence, k=8, c(n) = 1 for all n >= 1, and C(x) = x/(1-x). Thus, a(n) = a_8(n) for all n >= 1. Since a_k(n) = 0  for 1 <= n <= k-1, the offset of this sequence is n = k = 8. Applying the formula for the g.f. of DIK[8] of (c(n): n >= 1) with C(x) = x/(1-x) and k=8, we get Herbert Kociemba's formula below.
Here, a(n) is defined to be the number of n-bead bracelets of two colors with 8 red beads and n-8 black beads. But it is also the number of dihedral compositions of n with 8 parts. (This statement is equivalent to Vladimir Shevelev's statement above that a(n) is the "number of non-equivalent necklaces of 8 beads each of them painted by one of n colors." By "necklaces", he means "turnover necklaces". See the second paragraph of Section 2 in his 2004 paper in the Indian Journal of Pure and Applied Mathematics.)
Two cyclic compositions of n (with k = 8 parts) belong to the same equivalence class corresponding to a dihedral composition of n if and only if one can be obtained from the other by a rotation or reversal of order. (End)

			From _Petros Hadjicostas_, Jul 14 2018: (Start)
Every n-bead bracelet of two colors such that 8 beads are red and n-8 are black can be transformed into a dihedral composition of n with 8 parts in the following way. Start with one R bead and go in one direction (say clockwise) until you reach the next R bead. Continue this process until you come back to the original R bead.
Let b_i be the number of beads from R bead i until you reach the last B bead before R bead i+1 (or R bead 1). Here, b_i = 1 iff there are no B beads between R bead i and R bead i+1 (or R bead 8 and R bead 1). Then b_1 + b_2 + ... + b_8 = n, and we get a dihedral composition of n. (Of course, b_2 + b_3 + ... + b_8 + b_1 and b_8 + b_7 + ... + b_1 belong to the same equivalence class of the dihedral composition b_1 + ... + b_8.)
For example, a(10) = 5, and we have the following bracelets with 8 R beads and 2 B beads. Next to the bracelets we list the corresponding dihedral compositions of n with k=8 parts (they must be viewed on a circle):
RRRRRRRRBB <-> 1+1+1+1+1+1+1+3
RRRRRRRBRB <-> 1+1+1+1+1+1+2+2
RRRRRRBRRB <-> 1+1+1+1+1+2+1+2
RRRRRBRRRB <-> 1+1+1+1+2+1+1+2
RRRRBRRRRB <-> 1+1+1+2+1+1+1+2
(End)

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.

Andrew Howroyd, Table of n, a(n) for n = 8..1000
Christian G. Bower, Transforms (2)
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)
Richard H. Reis, A formula for C(T) in Gupta's paper, Indian J. Pure and Appl. Math., 10 (1979), no. 8, 1000-1001.
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]
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 (4,-4,-4,11,-8,0,8,-10,0,8,0,-10,8,0,-8,11,-4,-4,4,-1).

Column k=8 of A052307.

Cf. A008805, A032193, A032279, A032280, A032281, A032282, A119963, A292906.

k = 8; 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=8;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 *)

S. J. Cyvin et al. (1997) give a g.f. (See equation (18) on p. 870 of their paper. Their g.f. is the same as the one given by V. Jovovic below except for the extra x^8.) - Petros Hadjicostas, Jul 14 2018
G.f.: (x^8/16)*(1/(1 - x)^8 + 4/(1 - x^8) + 5/(1 - x^2)^4 + 2/(1 - x^4)^2 + 4/(1 - x)^2/(1 - x^2)^3) = x^8*(2*x^10 - 3*x^9 + 7*x^8 - 6*x^7 + 7*x^6 - 2*x^5 + 2*x^4 - 2*x^3 + 5*x^2 - 3*x + 1)/(1 - x)^8/(1 + x)^4/(1 + x^2)^2/(1 + x^4). - Vladeta Jovovic, Jul 17 2002
a(n) = ((n+4)/32)*s(n,0,8) + ((n-4)/32)*s(n,4,8) + (48*C(n-1,7) + (n+1)*(n-2)*(n-4)*(n-6))/768, if n is even >= 8; a(n) = (48*C(n-1,7) + (n-1)*(n-3)*(n-5)*(n-7))/768, if n odd >= 8, where s(n,k,d)=1, if n == k (mod d), and 0 otherwise. - Vladimir Shevelev, Apr 23 2011
G.f.: k=8, x^k*((1/k)*Sum_{d|k} phi(d)*(1-x^d)^(-k/d) + (1+x)/(1-x^2)^floor((k+2)/2))/2. - Herbert Kociemba, Nov 05 2016 [edited by Petros Hadjicostas, Jul 18 2018]
From Petros Hadjicostas, Jul 14 2018: (Start)
a(n) = (A032193(n) + A119963(n, 8))/2 = (A032193(n) + C(floor(n/2), 4))/2 for n >= 8.
The sequence (a(n): n >= 8) is the output sequence of Bower's "DIK[ 8 ]" (bracelet, indistinct, unlabeled, 8 parts) transform of 1, 1, 1, 1, ...
(End)

Sequence extended and description corrected by Christian G. Bower

Name edited by Petros Hadjicostas, Jul 20 2018

A143654 Array T(n,k) read by rows: number of binary bracelets with n beads, k of them 0, with 00 prohibited, (n >= 2, 0 <= k <= floor(n/2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 4, 3, 1, 1, 1, 4, 5, 4, 1, 1, 1, 5, 7, 8, 3, 1, 1, 1, 5, 8, 10, 5, 1, 1, 1, 6, 10, 16, 10, 4, 1, 1, 1, 6, 12, 20, 16, 7, 1, 1, 1, 7, 14, 29, 26, 16, 4, 1, 1, 1, 7, 16, 35, 38, 26, 8, 1, 1, 1, 8, 19, 47, 57, 50

Offset: 2

Washington Bomfim, Aug 28 2008

The initial columns give A057427, A057427, A004526, A069905, A005232, A032279, A005513, A032280, A005514, A032281, A005515, A032282, A005516. Row sums give A129526.
A binary bracelet with n beads, k of them 0, with 00 prohibited has from 0 to floor(n/2) beads 0, i.e., 0 <= k <= floor(n/2). If n is even, the bracelet 0101...01 with n/2 beads of each kind does not have 00 and we cannot change any 1 of it to a 0. If n is odd we cannot change a 1 to a 0 in the bracelet 0101...011 with (n-1)/2 beads 0.
The number of binary bracelets with n beads, 0 <= k <= floor(n/2) of them 0 with 00 prohibited, is equal to the number of binary bracelets with n-k beads, k of them 0. See below.
Let B be a binary bracelet with n-k beads, k of them 0. If we insert one 1 (circularly) after a 0 of B, we obtain a bracelet with n-k+1 beads, k of them 0.
If we do this insertion k times, each time after a distinct 0 of B, we obtain a bracelet with n = n-k+k beads, k of them 0, with 00 prohibited.
On the contrary, Let B be a binary bracelet with n beads, k of them 0, with 00 prohibited. If we remove from B one 1 that is after a 0, we obtain a bracelet of n-1 beads, k of them 0. (If not and we undo the removal, the configuration obtained cannot be a bracelet and this is absurd.) If we repeat this removal k times, after each distinct bead 0, we obtain a bracelet with n-k beads, k of them 0.

			Array begins
1 1
1 1
1 1 1
1 1 1
1 1 2 1
1 1 2 1
1 1 3 2 1
1 1 3 3 1
1 1 4 4 3 1
...
A129526(10) = A057427(10) + A057427(9) + A004526(8) + A069905(7) + A005232(6) +
A032279(5) = 1+1+4+4+3+1 = 14.

Cf. A057427, A004526, A069905, A005232, A032279, A005513, A032280, A005514, A032281, A005515, A032282, A005516. Row sums of array give A129526.

cp's OEIS Frontend

A052307 Triangle read by rows: T(n,k) = number of bracelets (reversible necklaces) with n beads, k of which are black and n - k are white.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A032279 Number of bracelets (turnover necklaces) of n beads of 2 colors, 5 of them black.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

A189976 a(n) is the number of incongruent two-color bracelets of n beads, 8 of them black (A005514), having a diameter of symmetry.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

A005514 Number of n-bead bracelets (turnover necklaces) with 8 red beads and n-8 black beads.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A143654 Array T(n,k) read by rows: number of binary bracelets with n beads, k of them 0, with 00 prohibited, (n >= 2, 0 <= k <= floor(n/2)).

Views

Author

Keywords

Comments

Examples

Crossrefs