TITLE: What is the proper way to extend the Fibonacci numbers to negative numbers?
QUESTION [32 upvotes]: I have a feeling this question is a duplicate, but it's not coming up in "Questions that may already have your answer."
We all know very well that $F(0) = 0$, $F(1) = 1$ and $F(n) = F(n - 2) + F(n - 1)$ for all $n > 1$.
I'm wondering about $n < 0$. My first thought was $F(-n) = -F(n)$, which is appealing from a multiplicative point of view, as it seems to preserve certain identities, like $F(n)^2 = F(n - 1) F(n + 1) - (-1)^n$.
But it doesn't quite make sense from an additive point of view, it doesn't seem to work both "forwards" and "backwards." For example, it would give us $F(-1) + F(0) = -1 \neq F(1)$.
How do we extend $F(n)$ to negative $n$ so as to maintain both the related identities and the basic defining identity?

REPLY [11 votes]: By now you know very well how to determine the Fibonacci numbers for negative indices, albeit by the recursion formula or the Binet formula as well as various others. My contribution is to show you what it looks like. The figure below is my crab-claw spiral; it consists of a Fibonacci spiral for $f_{-32}:f_{32}$. The image on the left is the complete spiral, that on the right is a detail at approximately $10^6$-fold zoom. The cusps are due to the alternating signs in the negative indices.