I have a 2-D square lattice and I am interested in finding the number of chains (series of squares) that I can surround the origin with. the length of the chain is from 4 to lets say to 10 squares. I have found this formula which upper bounds the number of chains $2*(z-2)*3^{z-2}$ where $z$ is the number of squares, the problem is that formula gives loose upper bounds. For example for $z=4$ it gives $2*2*9=36$ ways while practically there is only 1 way. Any help, guidance will be appreciated.
Regards.
Not an answer, but too long for a comment. It is almost done specifying what we are counting. In the figure below, the first shape is the only path with length $8$. The next three have length $10$. Are the last two allowed, with a square that touches three others? Are the last two the same, as they are reflections of each other. As we count rotations the same, I suggest we count reflections the same as well. In that case there are two paths of length $10$. I have found $10$ length $12$ paths, shown in the last two lines. I suspect it is not complete. I believe a hand (or computer) count of length $14$ would suffice to see if the sequence is in OEIS, but this is not enough-it yields $182$ pages.