In number theory, difference tables can be constructed to develop and visualize patterns within a given sequence that can sometimes lead to constructing a closed form representation of the sequence. Analyzing patterns in difference tables and corresponding diagonals for various classes of functions, including exponential functions and combinations of these exponentials, reveals a significant amount of structure. This paper serves to explain some of the most notable patterns along diagonals of difference tables and difference tables formed from the diagonals of the previous difference tables of a given sequence of numbers. These diagonals, which can be manipulated to find closed forms for functions of the form f(n)=nkjn where k and j are natural numbers by using a corresponding leading coefficient on a polynomial equivalent to the (j-1)th diagonal, can be found with a simple algorithm.

Author Bio

Dalton completed this research as part of a number theory and a thesis class at The University of Montana Western in Dillon, Montana in the fall of 2015 and spring of 2016. He grew up in Montana, and is accustomed to a very rural lifestyle. He loves to hike and fish and be outside. He has a deep passion for mathematics, martial arts, music, and learning in general. He is currently teaching high school mathematics in a small town in northern Montana. He plans to continue exploring his interests in mathematics to wherever it may take him. It is his hope that this paper on patterns, difference tables, and Pascal’s Triangle along with original theoretical methods for constructing closed forms of a variety of complex sequences will be intriguing to a wide audience of mathematicians.