Problem:

We place labeled points on a circle as follows. At step 1, take two points at opposite ends of a diameter and label them both $1$. At step $n > 1$, place a point at the midpoint of each arc created at step $n - 1$ and label it with the sum of the labels at the two adjacent points. What is the total sum of the labels after step $n$?

For example, after step 4 we have: $1$, $4$, $3$, $5$, $2$, $5$, $3$, $4$, $1$, $4$, $3$, $5$, $2$, $5$, $4$, $3$, $4$.