A005648 Number of 2n-bead black-white reversible necklaces with n black beads. (Formerly M0878)

1, 1, 2, 3, 8, 16, 50, 133, 440, 1387, 4752, 16159, 56822, 200474, 718146, 2587018, 9398520, 34324174, 126068558, 465093571, 1723176308, 6407924300, 23910576230, 89494164973, 335913918902, 1264107416466

Offset

0,3

Comments

a(n) is the coefficient of c_1^n*c_2^n in the cycle index polynomial for the dihedral group D_{2*n} evaluated with the figure counting polynomial c = c_1 + c_2, n>=1, abbreviated as Z(D_{2*n},c). See, e.g., the Harary-Palmer reference (given under A212355), p. 42, Theorem (PET), and the example for all 6 two-colored 4-bracelets (called there necklaces) on p. 44, Figure 2.4.2. - Wolfdieter Lang, Jun 05 2012

References

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1665 (terms 0..200 from Andrew Howroyd)

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

Paul Melotti, Sanjay Ramassamy, Paul Thévenin, Points and lines configurations for perpendicular bisectors of convex cyclic polygons, arXiv:2003.11006 [math.CO], 2020.

E. M. Palmer and R. W. Robinson, Enumeration of self-dual configurations Pacific J. Math., 110 (1984), 203-221.

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

Formula

a(n) = ( Sum_{d|n} phi(n/d)*C(2*d, d) )/(4*n) + C(2*k, k)/2, where k = floor(n/2). - Michael Somos

a(n) = (A003239(n) + C(2*k, k))/2, where k = [ n/2 ]. - R. J. Fletcher, (yylee(AT)mail.ncku.edu.tw)

Example

a(2) = 2: BBWW, BWBW.
a(3) = 3: BBBWWW, BBWBWW, BWBWBW.
a(4) = 8: BBBBWWWW, BBBWBWWW, BBBWWBWW, BBWWBBWW, BBWBWBWW, BBWBWWBW, BBWBBWWW, BWBWBWBW.

Mathematica

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], {n, 27}] (* Robert G. Wilson v, Mar 29 2006 *)
a[0] = 1; a[n_] := 1/2*(Binomial[2*Quotient[n, 2], Quotient[n, 2]] + DivisorSum[n, EulerPhi[#]*Binomial[2*n/#, n/#]&]/(2*n)); Array[a, 26, 0] (* Jean-François Alcover, Nov 05 2017, translated from PARI *)

Prog

(PARI) a(n) = 1/2*( binomial(2*(n\2), n\2) + if(n<1, n >= 0, sumdiv(n, k, eulerphi(k)*binomial(2*n/k, n/k))/(2*n) ));

Crossrefs

Cf. A000984, A003239.

Sequence in context: A204516 A331679 A277346 * A113947 A102008 A200083

Adjacent sequences: A005645 A005646 A005647 * A005649 A005650 A005651

Keyword

nonn,nice,easy

Author

N. J. A. Sloane

Extensions

Sequence extended and description corrected by Christian G. Bower

Example n=8 (word no. 6) corrected by Wolfdieter Lang, Jun 05 2012

Status

approved