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

1,3

Comments

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

References

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

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

Links

Table of n, a(n) for n=1..37.

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

Formula

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

Example

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

Mathematica

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

Prog

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

Crossrefs

Column 3 of A330618.

Cf. A000031, A093368.

Sequence in context: A274621 A291725 A003479 * A054186 A347212 A032156

Adjacent sequences: A093364 A093365 A093366 * A093368 A093369 A093370

Keyword

nonn

Author

N. J. A. Sloane, Apr 28 2004

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

Status

approved