I am assuming that $F$ is a function from some set back to itself, so that $F^n$ stands for $F\circ F\circ\dots\circ F$ , the n-fold composition, and that your $\emptyset$ thing is the $0$ vector. The subscript $n$ indicates that $F^n(0)$ is the $n$-th term of the sequence. (Under these assumptions, if $F$ linear then the sequence is trivial)
In general, a sequence in a set $A$ is just a function $s:\mathbb{N}\rightarrow A$, where $s(n)$ is viewed as the n-th term of the sequence. Often we see $\{s(n)\}_n$ or $\{s(n)\}_{n\in\mathbb{N}}$ denoting the image of $s$, indicating that this image is parametrized by $n\in\mathbb{N}$. In your case, $s(n)=F^n(0)$.
As another example, if we want a concise way of writing the sequence of rational numbers $\frac{1}{n}$ as $n$ ranges over $\mathbb{N}$, we could write $\{\frac{1}{n}\}_n$