Did you know that the nth Fibonacci number is $$F_n = \frac{1}{\sqrt{5}} \left[ \left(\frac{1 + \sqrt{5}}{2}\right)^n - \left(\frac{1-\sqrt{5}}{2}\right)^n \right]$$? What does this have to do with the infinite polynomial $$f(x) = 1 + x + 2x^2 + 3x^3 + 5x^4 + 8x^5 + \cdots$$? Generating functions are a beautiful tool for messing with sequences and figuring out recurrence relations. Armed with these and some arithmetic, we can come up with formulas like these with ease!