Fibonacci
Perrin en Fibonacci
On the previous page we saw some peculiar properties of the polynomials
x2 - x - 1 and x3 - x - 1.
On this page we will investigate uniformity and incongruity between these polynomials.
First: the largest root of the polynomials is respectively
= 
(1 + (1 + (1 + ...)))
and 
= 
(1 + (1 + (1 + ...))). (see Numerical application Fibonacci)
The sequence of numbers defined by P1 = 0, P2 = 2, P3 = 3 and
Pm+3 = Pm+1 + Pm is called the sequence of Perrin.
The fractions P2/P1,P3/P2,P4/P3,... approach the
unique real zero of the equation x3 - x - 1 = 0 (see Numerical application Fibonacci).
As you know (see Numerical application Fibonacci) the sequence F2/F1,F3/F2,F4/F3,... approaches the zero of the equation x2 - x - 1 = 0.
The sequence of Perrin also satisfies the recursion Pm+5 = Pm+4 + Pm (which is easy to proof).
The numbers in the sequence of Perrin have the following property:
If n is prime, then n divides Pn.
The converse is not always true, i.e.
if n is not prime, then Pn is not necessarily a prime, although this seems to be rare (q=271441 is the smallest non-prime for which q divides Pq).
A sequence of squares, whose sides have lengths of consecutive Fibonacci numbers, can be folded spirally (see the left picture).
The same is true for a sequence of equilateral trangles, with sidelengths equal to the lengths of consecutive Perrin numbers (see the right picture).
The triangle with sides of length P3 (we will abbreviate this by saying: triangle P3) overlaps triangle P5.
(In order to show that the lengths of the triangles are indeed Perrin numbers,
we notice that we can reflect a triangle Pm along one of its sides such, that
one of its sides coincides with (the continuation) of a side of triangle Pm+1,
and that those two sides are parallel to (and just as long as) one side of triangle Pm+3).
When we add the areas of the squares we obtain the formula:
F12 + F22 + F32 + ... + Fn2 = FnFn+1
For the triangles we have a similar formula:
P12 + P22 + P32 + ... + Pn2 = Pn+22 - Pn-12 - Pn-32 - 1 = 2PnPn+1 - Pn-22 - 1.