Re-formation of Fibonacci sequence formula

Question

I was doing some question on Fibonacci Sequence and came across this formula $$\large F_n = \frac{\left(\frac{1+\sqrt5}{2}\right)^n - \left(\frac{1-\sqrt5}{2}\right)^n}{\sqrt5}$$ I started thinking “Is there a better way to write this formula? Because this seems ugly, look at the denominator, it has surd and the numerator is also not good for easy calculation”
So I did some calculations and reformed it in another way $$ \large F_n^2 = \left\{ \begin{array}{ll} \dfrac{\left\lfloor \left(\frac{3+\sqrt5}{2}\right)^n \right\rfloor-1} {5}; \;\text{n is even}\\\\ \dfrac{\left\lfloor \left(\frac{3+\sqrt5}{2}\right)^n \right\rfloor+3}{5} ; \;\text{n is odd} \end{array} \right. $$ How can we further simplify it? I want to cancel $5$ in the denominator and need to prove that these two expressions are perfect squares