Number of ways to dissect a square into triangles of equal area

Monsky's theorem states that it is impossible to dissect a square into an odd number of triangles of equal area. If $n$ is an even integer, I am interested in the number of ways of dissecting a square into $n$ such triangles. Call $s_n$ this number. My question is:

What are the first terms of the sequence $s_n$?

with an option on this other question:

How can we compute $s_n$ for a given $n$?

Judging from the lack of answers to this question, it does not seem trivial that these numbers are even finite, let alone known. Obviously, $s_2=2$, and I believe that $s_4=25$. Other values are unknown to me.

Added 15/05/2015: I believe that $s_6$ is at least (and perhaps exactly) equal to $818$. This, however, does not lead to any sequence of the OEIS, so I'd be happy to be proved wrong.

Added 03/06/2015: For $n=2,4$ (but not for $n=6$, see mjqxxxx's answer below) the equidissections of the square that I could find all involve triangles whose vertices are rational numbers. For larger $n$, it is not sufficient to look only at equidissections satisfying this property.

Solution 1:

Even for $n=6$ it doesn't seem necessary for the coordinates to be rational. For instance, consider the following dissection into six right triangles of equal area:

The four interior points do not, I think, have rational coordinates. Calling the lower left corner $(0,0)$ and the lower right corner $(1,0)$, I found the interior point at lower right to be at $\left(\frac{1}{2} + \frac{1}{6}\sqrt{5}, \frac{1}{3}\right)$.