I was reading this paper, and came across an equation which gives an expression for the number of necklaces of beads in two colors, with length n. $Z_n = \dfrac{1}{n} \displaystyle \sum \limits_{d \mid n} \phi \left( d \right) 2^{n/d}$ The author says that this equality is well known, and cites Golomb and Riordan as […]

I’ve read that one can use the Polya enumeration theorem or the Burnside’s lemma to count the number of necklaces using $n$ beads from $k$ colors. Can we then find a way to count the number of necklaces such that no adjacent beads are of the same color?

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