We know that $ (2+\sqrt{3})^n + (2-\sqrt{3})^n$ is an integer (See hear).

However, we want to write the formula \begin{align} &\frac{3+\sqrt{3}}{6} (2+\sqrt{3})^n + \frac{3-\sqrt{3}}{6} (2-\sqrt{3})^n\ &=\frac{1}{6} \left[(2+\sqrt{3})^{n+1} + (2-\sqrt{3})^{n+1} + (2+\sqrt{3})^n + (2-\sqrt{3})^n\right] \end{align} to the form $ $ a^2 + 2\,b^2,\ (a, b \in \mathbb{N}).$ $

How?