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Search: id:A000011
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| A000011 |
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Number of n-bead necklaces (turning over is allowed) where complements are equivalent.
(Formerly M0312 N0114)
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+0
17
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| 1, 1, 2, 2, 4, 4, 8, 9, 18, 23, 44, 63, 122, 190, 362, 612, 1162, 2056, 3914, 7155, 13648, 25482, 48734, 92205, 176906, 337594, 649532, 1246863, 2405236, 4636390, 8964800, 17334801, 33588234, 65108062, 126390032, 245492244, 477353376, 928772650, 1808676326, 3524337980
(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. 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.
W. D. Hoskins; Anne Penfold Street, Twills on a given number of harnesses, J. Austral. Math. Soc. Ser. A 33 (1982), no. 1, 1-15.
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LINKS
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T. D. Noe, Table of n, a(n) for n = 0..200
Joerg Arndt, Fxtbook
H. Bottomley, Initial terms of A000011 and A000013
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
Index entries for sequences related to necklaces
Index entries for sequences related to bracelets
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FORMULA
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(A000013(n)+2^[n/2])/2.
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MAPLE
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with(numtheory): A000011 := proc(n) local s, d; if n = 0 then RETURN(1) else s := 2^(floor(n/2)); for d in divisors(n) do s := s+(phi(2*d)*2^(n/d))/(2*n); od; RETURN(s/2); fi; end;
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MATHEMATICA
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a[n_] := Fold[ #1 + EulerPhi[2#2]2^(n/#2)/(2n) &, 2^Floor[n/2], Divisors[n]]/2
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PROGRAM
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(PARI) a(n)=if(n<1, !n, 2^(n\2)/2+sumdiv(n, k, eulerphi(2*k)*2^(n/k))/n/4)
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CROSSREFS
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Cf. A000013. Bisections give A000117 and A092668.
Sequence in context: A016116 A060546 A120803 this_sequence A022476 A000013 A064484
Adjacent sequences: A000008 A000009 A000010 this_sequence A000012 A000013 A000014
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KEYWORD
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nonn,nice,easy
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AUTHOR
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njas
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EXTENSIONS
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Better description from Christian G. Bower (bowerc(AT)usa.net). More terms from David W. Wilson (davidwwilson(AT)comcast.net), Jan 13 2000.
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