SIMON-MARAIS 2017 B4

The following problem is open in the sense that no solution is currently known. Progress on the problem may be awarded points. An example of progress on the problem is a non-trivial bound on the sequence defined below.

For each integer $n \geq 2$, consider a regular polygon with $2n$ sides, all of length 1. Let $C(n)$ denote the number of ways to tile this polygon using quadrilaterals whose sides all have length 1.

Determine the limit inferior and the limit superior of the sequence defined by

$$\frac{1}{n^2} \log_2 C(n).$$