Let's create an arithmetic progression that has the property that the sum of its first $n$ terms is $4 n^{2}$.
According to the task,

$$
\frac{2 a+(n-1) d}{2} \cdot n=4 n^{2}
$$

from which

$$
n(d-8)+2 a-d=0
$$

The expression on the left side can only be 0 for all values of $n$ if

$$
d-8=0 \quad \text{and} \quad 2 a-d=0
$$

From these two equations,

$$
d=8 \quad \text{and} \quad a=4
$$

Therefore, the sought arithmetic progression is

$$
4,12,20, \ldots, 4+(n-1) 8
$$