To jump right into the rawness of my topic, how many addition expressions create a certain sum?
Like, for example, if $n = 3$, a formula would return $4$, because the possible addition equations with a sum of $n$ are (with only whole numbers) $1+2$, $2+1$, $3+0$, and $0+3$, which means there are four possible equations with a sum of $n$. Is there a formula for this?
If order matters and $0$ is allowed, so $0+3$ and $3+0$ both count, you are talking about weak compositions. For weak compositions into two parts there are $n+1$ of them. The first number ranges from $0$ through $n$ and the other is what is left.