Pell Numbers

In this paper, we find all the solutions of the Diophantine equation P +P m +P n = 2 a , in nonnegative integer variables (n, m, , a) where P k is the k-th term of the Pell sequence {P n } n≥0 given by P 0 = 0, P 1 = 1 and P n+1 = 2P n +P n−1 for all n ≥ 1. MSC: 11D45, 11B39; 11A25

Let $w_{n+2}=pw_{n+1}+qw_{n}$ for $n\geq0$ with $w_0=a$ and $w_1=b$. In this paper we find an explicit expression, in terms of determinants, for $\sum_{n\geq0} w_n^kx^n$ for any $k\geq1$. As a consequence, we derive all the previously known results for this kind of problems, as well as many new results.