Let  $p, q$ be integers and let $\alpha , \beta$ be the roots of the equation ${\mathrm{x}}^2-\mathrm{x}-1=0$, where $\mathrm{\alpha }\ne \mathrm{\beta }$.
For $\mathrm{n}=0,1,2,......$ let $a_{\mathit{n}}\mathit{=}p{\alpha }^{\mathit{n}}\mathit{+}q{\beta }^{\mathit{n}}$
Fact: If  $a \& b$ are rational numbers and $\mathrm{a}+\mathrm{b}\sqrt{5}=0$ and $a=0=\mathrm{b}$.