A generalised recurrence relation for irrational powers

In previous issues of Parabola, Randell Heyman showed that $c_n = (1 + \sqrt{2})^n + (1 - \sqrt{2})^n$ is an integer for each natural number $n$ and Xiaoyan Hu derived a recursion relation for this sequence. This article extends upon these results, by providing another method to arrive at the recursive relation. An application to finding solutions to Pell's Equation is given.