Problem 594

Rhombus Tilings

For a polygon $P$, let $t(P)$ be the number of ways in which $P$ can be tiled using rhombi and squares with edge length 1. Distinct rotations and reflections are counted as separate tilings.

For example, if $O$ is a regular octagon with edge length 1, then $t(O) = 8$. As it happens, all these 8 tilings are rotations of one another:

Let $O_{a,b}$ be the equal-angled convex octagon whose edges alternate in length between $a$ and $b$.
For example, here is $O_{2,1}$, with one of its tilings:

You are given that $t(O_{1,1})=8$, $t(O_{2,1})=76$ and $t(O_{3,2})=456572$.

Find $t(O_{4,2})$.

菱形地砖

对于多边形$P$，记$t(P)$为用边长为1的菱形和正方形地砖铺满$P$的方法数。旋转和翻转后相同的铺法视为不同的铺法。

例如，如果用$O$表示一个边长为1的正八边形，那么$t(O) = 8$。事实上，这8种铺法都可以通过旋转相互得到：

用$O_{a,b}$表示边长交替为$a$和$b$的等角凸八边形。
例如，如下是$O_{2,1}$和其中一种铺法：

已知$t(O_{1,1})=8$，$t(O_{2,1})=76$，以及$t(O_{3,2})=456572$。

求$t(O_{4,2})$。