The conventional Fibonacci sequence is given by
z(n+2)=z(n+1)+z(n), where z(0) and z(1) can be arbitrary complex
numbers. A quadratic Fibonacci sequence is given by
z(n+2)=Q(z(n),z(n+1)), where Q(w,z) is a polynomial of degree two in w
and z, i.e., Q(w,z)=az^2 +2bwz +cz^2 +dw +ez +f. Understanding such
sequences is equivalent to understanding the behaviour of iterates of
the map F(w,z)=(z,Q(w,z)), where w and z are complex numbers. A
special case is provided by the map F(w,z)=(z, z^2 +bw +c), which
gives a class of maps equivalent to the famous Henon maps. In this
talk we shall say a few words about general maps like F and then
concentrate on two examples: (a) G(w,z)=(z,w^2 +z) and
(b) H(w,z)=(z,z^2 +w). Despite their apparent simplicity, understanding
the behaviour of iterates of G and H poses a variety of intriguing
problems. We shall describe some theorems (e.g., G has periodic points
of every period and the closure of the set of periodic points is
compact), and we shall mention some open questions.