Question

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:

Answer 1

Answer:

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...