how to derive from fibonacci to negafibonacci? algebra question

Question

I was reading wiki on fibonacci sequence and saw $F_{-n} = (-1)^{(n+1)}F_n$ is called the negafibonacci sequence. How to get there from $F_n = F_{n-1} + F_{n-2}$ ??

Answer 1

We know $F_n$ for $n \geq 0$ (given that $F_0 = 0$). We also want to presevre the property that $F_n = F_{n-1}+F_{n-2}$. Ths, to get $F_n$ for negative $n$, we use the relation $F_n = F_{n+2}-F_{n+1}$. Now let's prove your equation using (strong) induction.

$n = -1$: We see that $F_{-1} = F_1 - F_0 = 1 = F_1(-1)^{-1+1}$

$n=-2$: $F_{-2} = F_{0}-F_{-1} = 0 - 1 = -1 = F_2(-1)^{-2+1}$

Assume the formula holds for some negative $n+1, n+2$, wehre $n < -2$. Then

$$F_n = F_{n+2} - F_{n+1} = (-1)^{n+3}F_{-n-2} - (-1)^{n+2}F_{-n-1} = $$

$$(-1)^{n+3}F_{-n-2} + (-1)^{n+3}F_{-n-1} = (-1)^{n+1}(F_{-n-2}+F_{-n-1}) = $$ $$(-1)^{n+1}F_{-n}$$