%I #13 Dec 22 2019 19:10:47

%S 0,0,2,3,6,11,18,33,58,105,186,349,630,1179,2190,4113,7710,14599,

%T 27594,52485,99878,190743,364722,699249,1342182,2581425,4971066,

%U 9587577,18512790,35792565,69273666,134219793,260301174,505294125,981706830,1908881897,3714566310

%N Number of n-bead necklaces using exactly three colors with no adjacent beads having the same color.

%C Original name: number of periodic cycles of iterative map described by Ma and Wainwright.

%D David W. Hobill and Scott MacDonald (zeened(AT)shaw.ca), Preprint, 2004.

%D P. K-H. Ma and Wainright, A dynamical systems approach to the oscillatory singularity in Bianchi cosmologies, Relativity Today, 1994.

%H P. K-H. Ma and Wainright, <a href="https://doi.org/10.1007/978-1-4757-9993-4_25">A dynamical systems approach to the oscillatory singularity in Bianchi cosmologies</a>, Deterministic Chaos in General Relativity, pp. 449-462, 1994.

%F a(n) = A000031(n) - (5 + (-1)^n)/2. - _Andrew Howroyd_, Dec 21 2019

%e a(3) = 2 because the two necklaces 123 and 132 have no adjacent equal elements. - _Andrew Howroyd_, Dec 21 2019

%t Table[Mod[n, 2] - 3 + DivisorSum[n, EulerPhi[n/#] 2^# &]/n, {n, 37}] (* _Michael De Vlieger_, Dec 22 2019 *)

%o (PARI) a(n)={n%2 - 3 + sumdiv(n, d, eulerphi(n/d)*2^d)/n} \\ _Andrew Howroyd_, Dec 21 2019

%Y Column 3 of A330618.

%Y Cf. A000031, A093368.

%K nonn

%O 1,3

%A _N. J. A. Sloane_, Apr 28 2004

%E Name changed by _Andrew Howroyd_, Dec 21 2019

%E a(1)-a(2) prepended and terms a(20) and beyond from _Andrew Howroyd_, Dec 21 2019