Use a tiling argument to give a combinatorial proof that \[F_n^2 + F_{n + 1}^2 = F_{2n + 1}.\] Also help with these too.
Use a tiling argument to give a combinatorial proof that \[F_n^2 + F_{n + 1}^2 = F_{2n + 1}.\]
We say that a set of integers is sparse if any two elements in the set differ by at least 3. Find the number of sparse subsets of $\{1, 2, 3, \dots, 12\}.$ (Note that the empty set is also sparse.)
Five people are passing a ball amongst themselves. The ball starts with Alonzo. Each person who has the ball passes it onto someone else. After the eighth pass, the ball returns to Alonzo. Find the number of possible sequences of passes.