  There exists a particular subset of algebraic power series over a finite
field which, for different reasons, can be compared to the subset of quadratic
real numbers. The continued fraction expansion for these elements, called
hyperquadratic, can sometimes be made explicit. Here we describe this expansion
for a wide family of hyperquadratic power series in odd characteristic. This
leads to consider interesting recurrent sequences in the finite base field when
it is not a prime field.
