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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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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; refs; listen; history; internal format)
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OFFSET
| 0,3
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REFERENCES
| 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.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| 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
| (A000013(n)+2^[n/2])/2.
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EXAMPLE
| Contribution from Jason Orendorff (jason.orendorff(AT)gmail.com), Jan 09 2009: (Start)
The binary bracelets for small n are:
. n: bracelets
. 0: (the empty bracelet)
. 1: 0
. 2: 00, 01
. 3: 000, 001
. 4: 0000, 0001, 0011, 0101
. 5: 00000, 00001, 00011, 00101
. 6: 000000, 000001, 000011, 000101, 000111, 001001, 001011, 010101 (End)
The above lines illustrate the fact that to get constant-width font in the Wiki version of the OEIS, you should begin each line with a dot.
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MAPLE
| 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
| a[n_] := Fold[ #1 + EulerPhi[2#2]2^(n/#2)/(2n) &, 2^Floor[n/2], Divisors[n]]/2
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PROG
| (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
| Cf. A000013. Bisections give A000117 and A092668.
Sequence in context: A163403 A183565 A120803 * A187213 A022476 A000013
Adjacent sequences: A000008 A000009 A000010 * A000012 A000013 A000014
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| 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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