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Say I have beads of $5$ colours, White, Yellow, Red, Green and Blue. I want to create a necklace of length $17$. How many unique necklaces can I create? (Up to rotation and reflection. For example, if the length was just 3, RGB and RBG should be identical)

The main problem I was facing was I could not choose a "uniform" factor to divide with. There are way too many different cases. Any tips/hints? I'm not even sure if this can be done via elementary combinatorics only.

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There is a fact sheet on necklaces and bracelets. We find the cycle index

$$Z(D_{17}) = \frac{1}{2} \frac{1}{17} \left( a_1^{17}+ 16 a_{17}\right) + \frac{1}{2} a_1 a_2^{8}.$$

We get for at most five colors,

$$\frac{1}{34} 5^{17} + \frac{16}{34} 5 + \frac{1}{2} 5^9.$$

This is $$\bbox[5px,border:2px solid #00A000]{ 22440372245.}$$

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