(EDITED)

Let $f(m,n)$ be the number of ways to tile a shape consisting of a $1 \times m$ rectangle on top of a $1 \times n$ rectangle with right sides aligned. You want $f(n,n)$.  If $m \ne n$, the leftmost tile could be a $1 \times j$ for
$1 \le j \le \max(m,n)$.  If $m = n$, you could have a $2 \times j$ or a $1 \times j$ atop a $1 \times k$.  Thus  
$$ f(m,n) = \cases{\sum_{j=1}^m f(m-j,n) & if $m > n$\cr
                   \sum_{j=1}^n f(m,n-j) & if $m < n$\cr
                   \sum_{j=1}^m f(m-j,m-j) + \sum_{j=1}^m \sum_{k=1}^m f(m-j,m-k) & if $m = n$\cr} $$
$f(n,n)$ is [OEIS sequence A034999](https://oeis.org/A034999).