Given a simple cubic graph with $n$ vertices (which implies that $n$ is even), what is a good upper bound on the number of cycles of length $n/2$ it can have?

A random cubic graph has $\Theta((4/3)^n/n)$ cycles of length $n/2$, if I did my sums right. So do random cubic bipartite graphs. Also the whole cycle space has size $2^{n/2+1}$, so twice that is a (silly) upper bound.

Where's the truth?

ADDED: Counts for n=4,6,...,24: 0,2,6,12,20,20,48,48,132,118,312 (not in OEIS). All these are unique except that for 20 vertices there are two graphs.