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?



Enumerating triangulations of quadrangulations in cycles
No triangulation occurs multiple times. At least one of the triangles $T$ of the triangulation has two edges on the boundary. There is a unique quadrant (I would say quadrilateral) $Q$ made up of two triangles of the triangulation, one of which is $T$. Remove the three edges of $T$, obtaining two quadrangulated cycles (possibly degenerate), and induct.


Note that not every triangulation of a $(2n+2)$-cycle is obtained by bisecting the quadrilaterals of a quadrangulation. Also, it is well-known that the number of quadrangulations of a $(2n+2)$-gon is $\frac{1}{2n+1}\binom{3n}{n}$. 


