Cutting 4 by $n$ grids into two congruent pieces

In the March 2025 issue of Pour la Science, Jean-Paul Delahaye described a wonderful solution to the following problem: How many ways can you divide a 3 by 2n rectangle into two connected, congruent pieces? We show that this problem can be solved by the transfer matrix method, and demonstrate this by computing the generating function for the number of ways to divide a 4 by n rectangle into two connected, congruent parts.