Consider a cycle of length $(2n+2)$. Now we quadrangulate this cycle into $n$ quadrants. We want to enumerate the number of quadrangulations, and we denote this number by $q_n$. Now we triangulate this quadrangulation by triangulating each quadrant. We denote the number of triangulations $t_n$. It is clear that $t_n = 2^nq_n$.
Do I count some triangulations multiple times? That is, will some triangulation be the result of multiple different quadrangulations?