A circular necklace contains n beads. Each bead is black or white. How many different necklaces can be made with n beads?

There is no clasp to identify a specific point on the chain, and a flipped over necklace is still the same necklace.
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To get you started:

With 1 bead, the necklace can be either 1 black or 1 white bead.

With 2 beads, the necklace can be either 2 black, 2 white, or 1 black-1 white

With 3 beads, the necklace can be either 3 black, 3 white, 2 black-1 white, 2 white-1 black, etc...

# Beads  Number of Necklaces
   1          2
   2          3
   3          4
   4          6
   5          8
   6         13

(In reply to re(5): There is an error is this puzzle by Richard)

Richard,

If you read that page carefully, I think you will find that the number 14 is not, in fact, correct.  It would be correct, if we don't respect flipping the necklace over, but the problem clearly states that you should.

The page refers to two similar, but different scenarios, as "fixed" and "free" necklaces--the latter able to be pulled out of the plane and "flipped over".  It is this latter scenario that corresponds to this problem.

Posted by Thalamus on 2004-08-20 08:10:10