Close packing of rectilinear polymers on a square lattice

The set of close packings of rectilinear $r$-mers on a square lattice is considered. It is shown that the number of configurations of $r$-reefs on a lattice containing $N$ sites increases with increasing $N$ not slower than $\exp{\{4GN/\pi r^2\} }$ and not faster than $(r/2)^{N/r^2}\exp{\{4GN/\pi r^2\} }$ if $r$ is even and
$$\biggl(\frac{r-1}{2}\biggr)^{N/r^2} \exp\biggl\{(N/\pi r^2)\int_0^{\pi} \operatorname{arch}\biggl(\frac{2r}{r-1}-\cos{\varphi}\biggr)\,d\varphi\biggr\},$$
if $r$ is odd ($G$ is Catalan's constant).