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Search: id:A005648
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| A005648 |
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Number of 2n-bead black-white reversible necklaces with n black beads.
(Formerly M0878)
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+0
7
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| 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
(list; graph; listen)
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OFFSET
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0,3
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Ascher, Marcia; Mu torere: an analysis of a Maori game. Math. Mag. 60 (1987), no. 2, 90-100.
E. M. Palmer and R. W. Robinson, Enumeration of self-dual configurations, Pacific J. Math., 110 (1984), 203-221.
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LINKS
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F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
Index entries for sequences related to bracelets
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FORMULA
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a(n) = (Sum_{d|n} (phi(n/d)*C(2d, d))/(2n) + C(2k, k))/2), where k = floor(n/2).
(A003239(n) + C(2k, k))/2, where k = [ n/2 ] (R. J. Fletcher, yylee(AT)mail.ncku.edu.tw).
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EXAMPLE
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a(2) = 2: BBWW, BWBW.
a(3) = 3: BBBWWW, BBWBWW, BWBWBW.
a(4) = 8: BBBBWWWW, BBBWBWWW, BBBWWBWW, BBWWBBWW, BBWBWBWW, BBWWBWBW, BBWBBWWW, BWBWBWBW.
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MATHEMATICA
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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 (russell(AT)post.harvard.edu), Sep 27 2004
Table[ f[n, 2n], {n, 27}] (from Robert G. Wilson v (rgwv(at)rgwv.com), Mar 29 2006)
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PROGRAM
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(PARI) a(n) = 1/2*( C(2*(n\2), n\2) + if(n<1, n >= 0, sumdiv(n, k, eulerphi(k)*C(2*n/k, n/k))/(2*n) )) where C(n, k)= if(k<0|k>n, 0, n!/k!/(n-k)!)
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CROSSREFS
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Cf. A000984, A003239. a(n)= (1/2) *(A000984([ n/2 ])+A003239(n)) (R. J. Fletcher, yylee(AT)mail.ncku.edu.tw).
Sequence in context: A011952 A032104 A051573 this_sequence A113947 A102008 A158921
Adjacent sequences: A005645 A005646 A005647 this_sequence A005649 A005650 A005651
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KEYWORD
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nonn,nice,easy
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AUTHOR
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N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
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Sequence extended and description corrected by Christian G. Bower (bowerc(AT)usa.net). Formula from Michael Somos.
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