The Fibonacci sequence can be extended backward to negative indices by rearranging the defining recurrence:? = ?+2 - ?+1. Here are the first several negative-index Fibonacci numbers:

f (n) = f (n+2) - f (n+1)

sequence: ...,-8,5,-3,2,-1,1,0,1,1,2,3,5,8, ...

Explanation:

Normal Fibonacci: f (n) = f (n-2) + f (n-1), sequence 0 1 1 2 3 5 8 ...

Now, replace n by n+2:

f (n+2) = f (n) + f (n+1)

and bring f (n) to the left while moving f (n+2) to the right:

f (n) = f (n+2) - f (n+1)

Now we can start applying it backwards.

f (0) = f (2) - f (1) = 0

f (-1) = f (1) - f (0) = 1

f (-2) = f (0) - f (-1) = - 1

f (-3) = f (-1) - f (-2) = 2

etc ...