I know this is elementary but bear with me.
Is there a way to calculate the number of diagonals an n-sided polygon has without using combinatorics???
I noticed that for every vertex there is an $\geq1$ other vertices to which it connects. For instance: In triangles it doesn't exist I guess.
In a quadrilateral $n=4$ a vertex connects to one other vertex.
A pentagon $n=5$. A vertex connects to $2$ other ones
Hexagon $n=6$: A vertex to $3$ other..
Can anything be made out of this, say a $1550$ sided polygon? Also what happens as $n\rightarrow\infty$