0,2

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).

N. J. A. Sloane, Table of n, a(n) for n = 0..300

Joerg Arndt, Fxtbook

H. Bottomley, Illustration of initial terms

P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.

F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

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

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.

Starting with "1" = binomial transform of [1, 3, 3, 1,...]; e.g. a(3) = 20 = (1, 3, 3, 1) dot (1, 3, 3, 1) = (1 + 9 + 9 + 1). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Oct 27 2007

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;

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]]

(PARI) a(n)=if(n<1, !n, (n%2+3)/4*2^(n\2)+sumdiv(n, d, eulerphi(n/d)*2^d)/2/n)

Cf. A001371 (primitive necklaces), A000031 (if cannot turn necklace over), A000011, A000013.

Adjacent sequences: A000026 A000027 A000028 this_sequence A000030 A000031 A000032

Sequence in context: A068597 A094372 A039880 this_sequence A155051 A018137 A084239

nonn,easy,nice,core

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Christian G. Bower (bowerc(AT)usa.net)