Among the sequences 
verifying a linear recurrence
, the sequence 
defined by the recurrence :
. In this
paper, a classification among these properties will be undertaken.
Part of them apply to all sequences , another
part depends on 
, a third part depends on properties of
(e.g. inversible, square, equal to 
) while quite none of
them appear to be really specific to the Fibonacci sequence itself.
The initial aim of the present study was to examine if formulae similar
to Eq. 2 can be found, for example a formula that
would give a linearization of 
.
To obtain the (negative) answer to this question, it has been useful
to reexamine Eq. 2 from the point of view of what
happens to other sequences, verifying other recurrences. Thereafter,
we have applied this fruitfull point of view to the period length
of the sequence .
The fact that at least 
for the recurrence 
explains why the conjecture
for all primes
[1] is so difficult to prove in the (genuine) Fibonacci
case.
The materials contained in this paper are organized as follows. Section 2
describe our notations and recall some elementary properties. In Section 3,
a "duplication formula" is stated that binds the
sequence of squared terms to the even and odd sub-sequences. Some
complements are given in Section 4. It will be
described how this duplication formula can be used to build a divide
and conquer algorithm that computes efficiently a given term of 
of such sequences. The special case of degenerate recurrences (
in our notations) is also considered.
The paper ends with Section 5 that examines how
the generating functions of the arithmetical subsequences, introduced
in Section 3, can be use to re-examine some
results concerning the period lenght of 
when assuming that 
and are coprime.
