BTW, on the subject of negative Fibonacci numbers, they do exist. Just reverse the recursion (F_n+F_{n+1} = F_{n+2}) => (F_{n-2} = F_n-F_{n-1}) and it works.
(Though they aren't terribly interesting: F_{-i} = -F_i * (-1)^i, so the negativeth Fibonacci number is either the negation of or equal to the positiveth Fibonacci number)