Triangulated n-gon How many ways are there to triangulate a regular convex n-gon, if two
triangulations are regarded as being the same if they can be made to coincide by a rotation of the polygon?
I was asked to express this in terms of the Catalan numbers and I got $G_n = \frac{C_{n-2}}{2} + \frac{C_{\frac{n}{2}-1}}{2} +\frac{2C_{\frac{n}{3}-1}}{3}$ where Cn is the Catalan number and if the subscript is not an integer than it is omitted. 
Now I have to try and prove that my expression $G_n$ is true but this is where I run into trouble. I tried induction and the inductive step was where it all fell apart. 
 A: Consulting  the  paper  by  Richard  Guy Dissecting  a  polygon  into
triangles and  using a somewhat  similar notation we  introduce $D_n$
which  is  $C_{n-2}$  where  $C_n$  is  the  Catalan  number  $$C_n  =
\frac{1}{n+1} {2n\choose n}$$ when $n$  is a positive integer at least
two  and zero  otherwise.  We  then have  consult  e.g. Wikipedia  on
Catalan   Numbers  that
the number of triangulations of the convex $n$-gon is given by $D_n.$
Now make  the following observation  concerning rotational symmetry
of  the  triangulations  of   the  regular  $n$-gon.   There  are  two
possibilities for the center of the $n$-gon, either it is located on a
diagonal  which is  in fact  a diameter  or it  is inside  one  of the
triangles. In the  first case we can have symmetry only  if we map the
diameter to itself by a $180$ degree rotation. Call the count of these
$F_n$.   In the  second case  we have  symmetry only  if  the triangle
containing  the center  is equilateral  and we  have two  rotations by
$120$ and $240$ degrees fixing  that triangle. Call the count of these
$R_n.$ Finally introduce $U_n$, triangulations having no symmetry. The
desired quantity is then given by
$$E_n = F_n + R_n + U_n.$$
We have by inspection that 
$$F_n = D_{n/2+1} \quad\text{and}\quad 
R_n = D_{n/3+1}.$$
Furthermore
$$D_n = \frac{n}{2} F_n + \frac{n}{3} R_n
+ n U_n$$
where we have computed the sizes of the respective orbits.
This yields
$$U_n = \frac{1}{n} D_n - \frac{1}{2} F_n - \frac{1}{3} R_n$$
so that 
$$E_n = \frac{1}{n} D_n + \frac{1}{2} F_n + \frac{2}{3} R_n$$
or
$$\bbox[5px,border:2px solid #00A000]{
E_n = \frac{1}{n} D_n 
+ \frac{1}{2} D_{n/2+1} + \frac{2}{3} D_{n/3+1}.}$$
This yields the sequence
$$1, 1, 1, 1, 4, 6, 19, 49, 150, 442,
\\ 1424, 4522, 14924, 49536, 167367, \ldots$$
which  points us  to  OEIS A001683,  where
detailed references await.  (We have started at index  $2$ rather than
$3$ to match the convention that the OEIS uses.)
