Suppose I want to form a 12-bead necklace such that each bead is either red or blue. Necklaces which differ by a rotation are considered the same and beads of the same color are indistinguishable. How may distinct necklaces can I make in this way?
My approach to this problem is as follows. We only have to consider the cases where the number of red beads is between $0$ and $6$, as the remaining cases are symmetric. If the number of red beads is $0,1,2$ the number of necklaces is easily found to be $1,1,6$, respectively. For $3$ to $6$ read beads the counting becomes more difficult. So lets suppose we have $k$ red beads (thus $12-k$ blue beads). Starting at some red bead and moving clockwise, let the number of blue beads between consecutive red beads be $x_1,x_2,...,x_k$ (these are non-negative integers). We need $x_1+...+x_k=12-k$ and thus the number of necklaces with $k$ read beads is the number of sets (recall the elements of a set are unordered) of $k$ nonegative integers with sum $12-k$, which we will call $f(k)$. So the answer to the problem is sum from $k=0$ to $5$ of $2f(k)$, plus $f(6)$. The problem is finding $f(k)$. Perhaps this is the wrong approach?