A093367 Number of n-bead necklaces using exactly three colors with no adjacent beads having the same color.

0, 0, 2, 3, 6, 11, 18, 33, 58, 105, 186, 349, 630, 1179, 2190, 4113, 7710, 14599, 27594, 52485, 99878, 190743, 364722, 699249, 1342182, 2581425, 4971066, 9587577, 18512790, 35792565, 69273666, 134219793, 260301174, 505294125, 981706830, 1908881897, 3714566310

Offset: 1

Author

N. J. A. Sloane, Apr 28 2004

Keywords

• nonn

Comments

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

Examples

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

References

• David W. Hobill and Scott MacDonald (zeened(AT)shaw.ca), Preprint, 2004.
• P. K.-H. Ma and J. Wainwright, A dynamical systems approach to the oscillatory singularity in Bianchi cosmologies, Relativity Today, 1994.

Links

• P. K.-H. Ma and J. Wainwright, A dynamical systems approach to the oscillatory singularity in Bianchi cosmologies, Deterministic Chaos in General Relativity, pp. 449-462, 1994.

Crossrefs

Column 3 of A330618.

Cf. A000031, A093368.

Programs

Mathematica

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

PARI

a(n)={n%2 - 3 + sumdiv(n, d, eulerphi(n/d)*2^d)/n}  \\ Andrew Howroyd, Dec 21 2019

Formula

a(n) = A000031(n) - (5 + (-1)^n)/2. - Andrew Howroyd, Dec 21 2019

Extensions

Name changed by Andrew Howroyd, Dec 21 2019

a(1)-a(2) prepended and terms a(20) and beyond from Andrew Howroyd, Dec 21 2019