OIM 2015 Problem 3

Let $\alpha$ and $\beta$ be the roots of $x^{2} - qx + 1$ , where $q$ is a rational number larger than $2$ . Let $s_1 = \alpha + \beta$ , $t_1 = 1$ , and for all integers $n \geq 2$ : $s_n = \alpha^n + \beta^n$ $t_n = s_{n-1} + 2s_{n-2} + \cdot \cdot \cdot + (n - 1)s_{1} + n$ Prove that, for all odd integers $n$ , $t_n$ is the square of a rational number.