Besides my internet muckings-about, I also try to do a bit of serious mathematics here and there. These are the first of what I hope to be many papers in my mathematical career. You'll need Adobe Reader to read these papers, but chances are your computer is already equipped with the proper software.
Combinatorial Properties of Billiards on an Equilateral Triangle
This is a slimmed-down version of my undergraduate math thesis at Millersville University. Imagine a ball bouncing about a pool table shaped like an equilateral triangle and say you want to bounce the ball around the walls a certain number of times before it returns its starting point (and would continue doing so indefinitely save for friction). You have created a periodic orbit of period n. The paper counts the number of periodic orbits of period n for any integer n.
k-Part Partitions of an Integer Congruent Modulo m
This is a work in progress that I hope to have ready for publication within the year. A k-part partition of an integer is a way of writing out the integer as a sum of k integers (e.g. 5 can be partitioned into 3 integers in two ways: 1+1+3,1+2+2 ). In this paper we allow zeros as valid addends (so 5 can be partitioned into 3 parts in five ways: 0+0+5, 0+1+4, 0+2+3, 1+1+3, 1+2+2). Furthermore, we add the restriction that the addends must all be congruent to each other (not necessarily to n itself) modulo some integer m. This means simply that the difference between any two of the addends must be a multiple of m (if we set m=2, only 1+1+3 meets all of the requirements for a 3-part partition of 5 congruent modulo 2). We want to count the number of partitions of n into k nonnegative integers (positives and zero) that are congruent to one another modulo m for any given k, n, and m. While the paper gives a method (known as a generating function) for any k, m, and n, it fails in giving all pertinent explicit formulas (something you can plug the values into and receive an answer)