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A000029
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Number of necklaces with n beads of 2 colors, allowing turning over.
(Formerly M0563 N0202)
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34
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1, 2, 3, 4, 6, 8, 13, 18, 30, 46, 78, 126, 224, 380, 687, 1224, 2250, 4112, 7685, 14310, 27012, 50964, 96909, 184410, 352698, 675188, 1296858, 2493726, 4806078, 9272780, 17920860, 34669602, 67159050, 130216124, 252745368, 490984488
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OFFSET
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0,2
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REFERENCES
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S. N. Ethier and J. Lee, Parrondo games with spatial dependence, Arxiv preprint arXiv:1202.2609, 2012. - From N. J. A. Sloane, Jun 10 2012
N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.
J. L. Fisher, Application-Oriented Algebra (1977) ISBN 0-7002-2504-8, circa p 215.
Martin Gardner, "New Mathematical Diversions from Scientific American" (Simon and Schuster, New York, 1966), pages 245-246.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
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).
A. M. Uludag, A. Zeytin and M. Durmus, Binary Quadratic Forms as Dessins, http://math.gsu.edu.tr/uludag/CHARKSANDDESSINS.pdf, 2012. - From N. J. A. Sloane, Dec 31 2012
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LINKS
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N. J. A. Sloane, Table of n, a(n) for n = 0..300
Joerg Arndt, Fxtbook, p.151
H. Bottomley, Illustration of initial terms
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
S. N. Ethier, Counting toroidal binary arrays, arXiv preprint arXiv:1301.2352, 2013.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
Eric Weisstein's World of Mathematics, Necklace
Eric Weisstein's World of Mathematics, e
Index entries for "core" sequences
Index entries for sequences related to bracelets
Index entries for sequences related to necklaces
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FORMULA
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Sum_{ d divides n } phi(d)*2^(n/d)/(2*n) + either 2^((n-1)/2) if n odd or 2^(n/2-1)+2^(n/2-2) if n even.
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MAPLE
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with(numtheory): A000029 := proc(n) local d, s; if n = 0 then RETURN(1); else if n mod 2 = 1 then s := 2^((n-1)/2) else s := 2^(n/2-2)+2^(n/2-1); fi; for d in divisors(n) do s := s+phi(d)*2^(n/d)/(2*n); od; RETURN(s); fi; end;
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MATHEMATICA
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a[0] := 1; a[n_] := Fold[ # 1 + EulerPhi[ # 2]2^(n/ # 2)/(2n) &, If[OddQ[n], 2^((n - 1)/2), 2^(n/2 - 1) + 2^(n/2 - 2)], Divisors[n]]
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PROG
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(PARI) a(n)=if(n<1, !n, (n%2+3)/4*2^(n\2)+sumdiv(n, d, eulerphi(n/d)*2^d)/2/n)
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CROSSREFS
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Row sums of triangle in A052307.
Cf. A001371 (primitive necklaces), A000031 (if cannot turn necklace over), A000011, A000013.
Cf. a(n) = A081720(n,2), n>=2. [From Wolfdieter Lang, Jun 03 2012]
Sequence in context: A068597 A094372 A039880 * A155051 A018137 A084239
Adjacent sequences: A000026 A000027 A000028 * A000030 A000031 A000032
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KEYWORD
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nonn,easy,nice,core
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AUTHOR
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N. J. A. Sloane.
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EXTENSIONS
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More terms from Christian G. Bower
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STATUS
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approved
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