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.
Solution
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