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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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97
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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
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OFFSET
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0,3
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COMMENTS
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a(n) is also the number of minimal fibrations of a bidirectional n-cycle over the 2-bouquet up to precompositions with automorphisms of the n-cycle and postcomposition with automorphisms of the 2-bouquet. (Boldi et al.) - Sebastiano Vigna, Jan 08 2018
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REFERENCES
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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
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Zhe Sun, T. Suenaga, P. Sarkar, S. Sato, M. Kotani, and H. Isobe, Stereoisomerism, crystal structures, and dynamics of belt-shaped cyclonaphthylenes, Proc. Nat. Acad. Sci. USA, vol. 113 no. 29, pp. 8109-8114, doi: 10.1073/pnas.1606530113.
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FORMULA
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a(n) = (A000013(n) + 2^floor(n/2))/2.
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EXAMPLE
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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)
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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
a[ n_] := If[ n < 1, Boole[n == 0], 2^Quotient[n, 2] / 2 + DivisorSum[ n, EulerPhi[2 #] 2^(n/#) &] / (4 n)]; (* Michael Somos, Dec 19 2014 *)
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PROG
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(PARI) {a(n) = if( n<1, n==0, 2^(n\2) / 2 + sumdiv(n, k, eulerphi(2*k) * 2^(n/k)) / (4*n))}; /* Michael Somos, Jun 03 2002 */
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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