Fillomino solutions -- are you crazy?!
Consider a polyomino \(S\). We want to divide \(S\) into several (may be one) smaller polyominoes. The division is called a Fillomino solution if no two polyominoes of equal size share a side. (They may touch at a point.) In other words, the division is a valid solution of a Fillomino puzzle.
Determine the number of distinct Fillomino solutions of the \(1 \times 100\) rectangle.