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It's very well known that the number of different necklaces using two colors of pebbles (Two necklaces which can be obtained from one another by rotation are considered the same), is exactly $\frac{1}{n}(\sum_{d|n} \phi(\frac{n}{d})2^d)$. What about the answer to this question:

How many different (as defined above) necklaces are there using exactly $m$ red and $n$ white pebbles?

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    At least if $\gcd(n.m)=1$, the answer is simply $\frac1{m+n}{m+n\choose n}$.2012-09-28
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    Yes, you're right. But things get different when they are not coprime.2012-09-28

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