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, 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)