9. (10 points) Place $n$ identical square pieces of paper on a table without overlapping, forming a multi-layer rectangular (including square) shape with at least two layers, and each upper square piece must have two vertices each on the midpoint of a side of a lower square piece. The figure below shows all different placement methods for $n=6$. How many different placement methods are there for $n=8$?
【Analysis】We can discuss various possible scenarios by layers, then sum them up. Since when $n=8$, the maximum number of layers for placement is 3, it is not difficult to find the total number of different placement methods.

【Solution】Solution: According to the analysis, we discuss based on the different number of layers:
(1) When the number of layers is 2, $7+1$ has 6 ways; $6+2$ has 4 ways; $5+3$ has 2 ways;
(2) When the number of layers is 3, $5+2+1$ has 3 ways; $4+3+1$ has 2 ways;

Therefore, there are a total of: $6+4+2+3+2=17$ ways.
The answer is: 17.