where 
, i.e.
will be referred as a "nothing but 
"
pattern, and a pattern of length 
where 
will be referred as an "any, one, one" pattern (the
sequence 
must be palindromic, since 
has to be so).
(from theorem 4.10).
When 
, the corresponding 
and 
are given by:
. Additionally, the parameter
must be even when 
. 
, where 
is the -th Fibonacci number. Thus the signatures
are given by:
, 
is also a pattern of
first kind and the parity condition over is here to select
the 
). The formulae
,
while the empty pattern occurs when 
.
, and 
can ever
be additionally required. In such a case, we have:
. 
, where 
is defined by 
and 
.
Thus the signatures are alternatively 
and 
and criterion
of theorem 4.10 (cor. 4.12)
is satisfied. Formulae
. Finally, 6.2
is an asymptotic formula, based upon
and 
found are quickly huge, and give no hint about the smallest
having a given period length.
Since 
,
we have 
when 
. In other words, when the 
-pattern
is "4", the -pattern is 
"1",
and the length of the period is exactly three times the length
of the period.
associated to a 
belong to the same pattern of first
kind 
, where
. And we have
where all 
are even and at least 
belongs to some 
such that 
is even, then all such share also
the above described as pattern of second kind.
and several matrices 
where
pattern follows from :
is a fixed point of 
when is a fixed point of 
. Conversely, if
is a fixed point of 
and as described, then
is a pattern, 
is odd and 
is a fixed
point of .
is defined by 
when
and, otherwise, by 
.
The pseudo period of is the period associated to ,
i.e one or three (ordinary) periods of . versus the pseudo-period of
for all 
in 
, we obtain
FIG. 1.1 (left : the whole figure; right:
the lower left corner zoomed in).
