Number of configurations for nRm are ...

nRm is quite similar to nCm, except that it is circular combination. I don't have a single formulae for that. But I have a solution.

nCm = sum of (n/fi * |(n/fi)Q(m/fi)|) -- (1)

|nRm| = sum of(|(n/fi)Q(m/fi)|) -- (2)

where fi is a gcd(n,m)=g including 1 and g.
nQm is the subset of configurations from nRm, s.t. nQm does not have any symmetric ones.

Proof:
The proof is by counting. So (2) is obvious. Only thing to remember is there for each element in Q there is one in R and viceversa.

Same type of counting works for (1) also.