The recurrence an=an−1+an−2a_n = a_{n-1} + a_{n-2}an=an−1+an−2 has roots ϕ=1+52\phi = \frac{1+\sqrt{5}}{2}ϕ=21+5 and ψ=1−52\psi = \frac{1-\sqrt{5}}{2}ψ=21−5. What is the closed form for a0=0,a1=1a_0=0, a_1=1a0=0,a1=1?
an=ϕn−ψn5a_n = \frac{\phi^n - \psi^n}{\sqrt{5}}an=5ϕn−ψn
an=ϕn+ψn5a_n = \frac{\phi^n + \psi^n}{\sqrt{5}}an=5ϕn+ψn
an=ϕn−ψna_n = \phi^n - \psi^nan=ϕn−ψn
an=ϕn−ψn2a_n = \frac{\phi^n - \psi^n}{2}an=2ϕn−ψn