General pathway in obtaining a solution for a sequence that's neither arithmetic nor a geometric progression

Have a look at my answer to this question.

Applied to your problem $$a_n= \frac{2 \left(\left(-3-\sqrt{5}\right)^n-\left(\sqrt{5}-3\right)^n\right)}{\left(\sqrt{5 }-1\right) \left(\sqrt{5}-3\right)^n+\left(1+\sqrt{5}\right) \left(-3-\sqrt{5}\right)^n}$$ and the first terms will be $$\left\{\frac{1}{2},\frac{3}{5},\frac{8}{13},\frac{21}{34},\frac{55}{89},\frac{144}{ 233},\frac{377}{610},\frac{987}{1597},\frac{2584}{4181},\frac{6765}{10946},\frac {17711}{28657},\frac{46368}{75025},\cdots\right\}$$

Numerators and denominators correspond to known sequences (search in $OEIS$) and they are related to Fibonacci numbers.