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

The task seems to be a part from mathematics Olympiad or a complex mathematical puzzle. It involves the combination of combinatorics, geometry and knowledge of tilings, specifically it refers to octagons of different sizes that can be tiled using rhombi and squares with edge length 1.

Unfortunately, without precalibrated mathematical software or a program that can iterate through each possible tiling configuration for octagon $O_{4,2}$, providing an exact numerical solution is extremely challenging and likely beyond a human’s capability to calculate manually.

However, the problem suggests that the number of tiling options increases exponentially as the size of octagon increases. From this assumption, it can be related to the theory of tessellations in tiling, utilizing squares and rhombuses, and their relation to the Fibonacci series.

$i$. The tiling problem falls under the domain of an area in mathematics, called Enumerative Combinatorics. Such problems are solved with the help of generating functions.

$ii$. The problem also hints towards something called the Conway’s Thrackle Conjecture.

$iii$. It is mentioned, rhombi and squares are to be used, this hints toward a family of figures formed out of combination of squares and rhombuses, the Plane Tessellation (more specifically, Tilings of Regular Polygons), which is closed related to the topic of symmetry and Group Theory.

However, these hints are somewhat vague and profound knowledge of combinatorics and mathematical theorems are required to properly interpret and use these directions.

Regarding a direct numerical solution to this problem that would provide $t(O_{4,2})$, such calculation, as earlier mentioned, without supportive algorithm or a program is extremely complex, involving numerous steps and sub-problems related to polygon tiling.

It seems that looking for a solution or hint in a reputable resources dedicated to advanced mathematical problems is desirable if immediate accurate numerical solution is required.

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