I need a formula to calculate the count of unique vertices for a subdivided triangle: $v = f(s)$ where $s$ is the count of subdivisions and $v$ is the count of vertices.
Let's take the biggest triangle in the picture. If I do not subdivide it I need $3$ vertices: $f(0) = 3$
If I subdivide it once I need 6 vertices: $f(1) = 6$
I counted the vertices for the next subdivisions since I am not able to create a formula for it:
$$f(2) = 15\\ f(3) = 45\\ f(4) = 153$$
Can someone help me to find $f$?
If I understood the question correctly, what you want is $$f(n)=\left(2^n+1\right)\left(2^{n-1}+1\right)$$ If you're wondering how it is obtained, I suggest you to read this.