Show that all terms of the sequence $a_n=\left(\frac{3+\sqrt{5}}{2}\right)^n+\left(\frac{3-\sqrt{5}}{2}\right)^n -2$ are integers. And when $n$ is even, $a_n$ can be expressed as $5m^2$, when $n$ is odd $a_n$ can be expressed as $m^2$.