Fibonacci-type Sequence with Complex Numbers I have been playing around with Fibonacci-type of sequence that involve complex numbers.  I have stumbled upon the following sequence, which seemed interesting to me:
$$0,1,2i,-3,-4i,5,6i,...$$
so $F_n = 2iF_{n-1} + F_{n-2}$.  These look like a sequence of natural numbers (except for $0$) where every other is multiplied by $i$ and the signs change after two sequences.  
I understand the algebra behind the above sequence, but I have been wondering whether there is an intuition behind why the sequence looks like a "modified" sequence of natural numbers.

The characteristic polynomial for your recursion is $$x^2-2ix-1=(x-i)^2$$
Visibly, this has a double root at $x=i$.  Thus the general form of the solution to the recursion is $$F_n=Ai^n+Bni^n$$  Using your initial conditions it is easy to specify the solution to your case.
