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for questions about sequences and series, e.g. convergence, closed form expressions, etc. Note that there is a different tag for spectral sequences, and also note that MathOverflow is not for homework. Please consider consulting the online encyclopedia for integer sequences, if you are trying to identify a given sequence that you have found in your research.
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Number and asymptotic for cyclic sequences
You are essentially counting necklaces, see e.g. http://theory.cs.uvic.ca/inf/neck/NecklaceInfo.html. The number of necklaces of length $l$ over a $k$-element alphabet is
$N_k(l)=\frac{1}{l}\sum_{d|l …