We have excellent approximations; for 19x19 the number is known to be approximately 2.081681994 * 10^170. Getting the exact number is the big challenge. You can

We have excellent approximations; for 19x19 the number is known to be approximately 2.081681994 * 10^170. Getting the exact number is the big challenge. You can see in the paper that we spent a lot of effort on reducing the complexity of the counting problem. We welcome suggestions for further improvement, but suspect our current algorithm (counting paths in the border state graph) is already close to optimal.