I am studying crystallography and came across the topic of voids. The voids are classified on the basis of the range of the radius ratio $r_{+}/r_{-}$ values obtained, where $r_{+}$ is the radius of the cation present in the voids and $r_{-}$ is the radius of the anions, under the case $r_{-}\gt r_{+}$. So there are certain terminology that tells the best fit and the max. fit or basically the lower limit and the upper limit of the radius ratio for which the cation of that radius can be accommodated within that type of void.
Let $a_n-1$ denote the upper limit on the radius ratio (or max. fit) that can be accommodated in the void with coordination number $n$.
$$\bbox[15px,border:2px solid #444444]{a_2=\frac{2}{\sqrt{3}}, a_3=\frac{\sqrt{3}}{\sqrt{2}},a_4=\sqrt{2}, a_6=\sqrt{3}, a_8=2}$$
For $n=2k, k\in\mathbb{Z^{+}}$, it seems that $a_{2k}-1=\sqrt{k}$, but that does not seem to be applicable to the $k=1$ case.
These radius ratios have been derived from geometry and trigonometry application on lattices as geometrical models. So, can this sequence be generalised for all $n$? Thanks