A185437 The least number of colors required to color an n-bead necklace so that each bead can be identified.

1, 2, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset

1,2

Comments

In other words, the least number of colors in any coloring of the necklace that is not symmetric under any element of the corresponding dihedral group.

Links

Table of n, a(n) for n=1..83.

Index entries for linear recurrences with constant coefficients, signature (1).

Formula

a(n) = 2 for all n > 5.

G.f.: x*(x^2+1)*(x^3-x-1)/(x-1). [Colin Barker, Oct 26 2012]

Example

For n=5, one coloring is ABBCC. Any coloring using two symbols will have two indistinguishable beads.
For n > 5, a coloring is ABAAB...B, where ... is zero or more B's. We can tell the A's apart because one has a B on either side, of the other two one is closer to the single B, and one is closer to the long sequence of B's. Of the B's, one has an A on either side. The remaining B's can be distinguished by counting along the string of B's starting at the end with a singleton A.

Crossrefs

Cf. A185436.

Sequence in context: A347824 A031283 A293229 * A335660 A210681 A366544

Adjacent sequences: A185434 A185435 A185436 * A185438 A185439 A185440

Keyword

nonn,easy

Author

Jack W Grahl, Jan 27 2011

Status

approved