A flea jumps on a regular $n$-gon A flea jumps on a regular $n$-gon with center $O$ with the following rules: it starts in $O$; if it's in $O$ it can jump to any vertex (with equal probability); if it's on a vertex it can jump in $O$ or to an adiacent vertex (with equal probability). Which is the probability that, after $n$ jumps, the flea is again in $O$?

**HINT:** The probability that the flea is at $O$ after $m$ rounds, can be computed recursively as $$p_m=\frac13(1-p_{m-1}). $$