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. [Cached copy, with permission, pdf format only]
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
KEYWORD
nonn,nice,easy
AUTHOR
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