The Necklace Problem

  1. How many different necklaces can be made with eight beads, where each bead may be either black or white, the beads being indistinguishable, except by color? The necklace is a complete circle, and a necklace with beads 1, 3 and 4 black would be identical to one with beads 2, 7 and 8 black.
  2. What if the necklace has 7 beads?
  3. 9 beads?
  4. 10 beads?
  5. Can this be generalized for N beads?

Source: Henry Ernest Dudeny's 536 Curious Problems & Puzzles, #458.

Solutions were received from many people, most having used a computer. The table of beads and necklaces is below: 
1              2 
2              3 
3              4 
4              6 
5              8 
6             13 
7             18 
8             30 
9             46 
10            78 
11           126 
12           224 
13           380 
14           687 
15          1224
The best solution discussion is actually already on the web at: The Necklace Problem at Eric's Treasure Trove of Mathematics.
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