The Necklace Problem

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.
 What if the necklace has 7 beads?
 9 beads?
 10 beads?
 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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