You're right it works fine.

Why would L(n) = L(n + 2) - L(n + 1) not work ?

example:

L(-1) = L(1) - L(0) = 

L(-1) = 1 - 2 = 

L(-1) = -1

another one:

L(-2) = L(0) - L(-1) = 

L(-2) = 2  -  -1  = 

L(-2) = 3

and so on ....

I would suggest using L(-n) = L(n)(-1)^n

I've modified the description,
can please check and if it's ok resolve the issue.