Enumerating solutions to grid based puzzles with a fixed number of rows

In this paper we demonstrate a method for counting the number of solutions to various logic puzzles. Specifically, we remove all of the "clues" from the puzzle which help the solver to a unique solution, and instead start from an empty grid. We then count the number of ways to fill in this empty grid to a valid solution. We fix the number of rows $r$, vary the number of columns $k$, and then compute the sequence $A_r(k)$, which gives the number of solutions on an empty grid of size $r \times k$.