A005648 - OEIS

%I M0878 #68 Jul 03 2020 06:23:11

%S 1,1,2,3,8,16,50,133,440,1387,4752,16159,56822,200474,718146,2587018,

%T 9398520,34324174,126068558,465093571,1723176308,6407924300,

%U 23910576230,89494164973,335913918902,1264107416466

%N Number of 2n-bead black-white reversible necklaces with n black beads.

%C 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

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

%H G. C. Greubel, <a href="/A005648/b005648.txt">Table of n, a(n) for n = 0..1665</a> (terms 0..200 from Andrew Howroyd)

%H Marcia Ascher, <a href="http://www.jstor.org/stable/2690304">Mu torere: an analysis of a Maori game</a>, Math. Mag. 60 (1987), no. 2, 90-100.

%H R. K. Guy & N. J. A. Sloane, <a href="/A005648/a005648.pdf">Correspondence, 1985</a>

%H Paul Melotti, Sanjay Ramassamy, Paul Thévenin, <a href="https://arxiv.org/abs/2003.11006">Points and lines configurations for perpendicular bisectors of convex cyclic polygons</a>, arXiv:2003.11006 [math.CO], 2020.

%H E. M. Palmer and R. W. Robinson, <a href="http://projecteuclid.org/euclid.pjm/1102711113">Enumeration of self-dual configurations</a> Pacific J. Math., 110 (1984), 203-221.

%H F. Ruskey, <a href="http://combos.org/necklace">Necklaces, Lyndon words, De Bruijn sequences, etc.</a>

%H F. Ruskey, <a href="/A000011/a000011.pdf">Necklaces, Lyndon words, De Bruijn sequences, etc.</a> [Cached copy, with permission, pdf format only]

%H <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>

%F 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_

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

%e a(2) = 2: BBWW, BWBW.

%e a(3) = 3: BBBWWW, BBWBWW, BWBWBW.

%e a(4) = 8: BBBBWWWW, BBBWBWWW, BBBWWBWW, BBWWBBWW, BBWBWBWW, BBWBWWBW, BBWBBWWW, BWBWBWBW.

%t 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 *)

%t Table[ f[n, 2n], {n, 27}] (* _Robert G. Wilson v_, Mar 29 2006 *)

%t 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 *)

%o (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) ));

%Y Cf. A000984, A003239.

%K nonn,nice,easy

%O 0,3

%A _N. J. A. Sloane_

%E Sequence extended and description corrected by _Christian G. Bower_

%E Example n=8 (word no. 6) corrected by _Wolfdieter Lang_, Jun 05 2012