Triangular Numbers and Combinatorics

The triangular number, \(T^2_n\), is found by counting the number of packed circles forming the equilateral triangle with \(n\) circles to a side. It is well-known that \(T^2_n = n(n+1)/2\), which happens to be the number of ways that 2 objects can be selected from among \(n+1\) objects. We explore this connection between combinatorics, the triangular numbers, and their higher-dimensional analogs.
Project Euler

Solutions and discussions of the Project Euler problems. These are a collection of increasingly challenging mathematical and computational exercises across the field of number theory. This is a work in progress and I will post new solutions as I solve them.