SymPy Gamma

rsolve (y(n+2)-y(n+1)-y(n),y(n),{y(0):0,y(1):1})

$$\begin{align}&\mathrm{Solve~the~recurrence~}- y{\left (n \right )} - y{\left (n + 1 \right )} + y{\left (n + 2 \right )} = 0\\&\mathrm{with~initial~conditions}\\&y{\left (1 \right )} = 1\\ &y{\left (0 \right )} = 0\end{align}$$

$$\frac{\sqrt{5}}{5} \left(\frac{1}{2} + \frac{\sqrt{5}}{2}\right)^{n} - \frac{\sqrt{5}}{5} \left(- \frac{\sqrt{5}}{2} + \frac{1}{2}\right)^{n}$$

2**(-n)*sqrt(5)*((1 + sqrt(5))**n - (-sqrt(5) + 1)**n)/5

$$\frac{2^{- n}}{5} \sqrt{5} \left(\left(1 + \sqrt{5}\right)^{n} - \left(- \sqrt{5} + 1\right)^{n}\right)$$