There are a lot of ways to define Catalan Numbers. We’ll number a few of them one by one. Diagonal-Avoiding Paths This is probably the interpretation that is easiest to get an implicit formula from. Take an \(n \times n\) grid, going from \((0, 0)\) to \((n, n)\), and consider... Continue reading...
Hey! I personally never know how to feel about social media. I have a few accounts and I browse content frequently, but I never share anything publicly. I feel that I’m at some middle ground between frequent social media users and non-users, and that interested me in getting perspectives from... Continue reading...
Decimals and Modular Arithmetic Let’s say we wanted to write \(0.\overline{315}\) as a fraction. Converting a repeating decimal to its fractional form usually follows the following process: Set \(x\) to equal your repeating decimal. If \(x\) is of length \(n\) digits, consider \(x * 10^n\). Subtracting \(x\) from this cancels... Continue reading...