fibonacci-numbers

The first 18 Fibonacci numbers denoted by for 0,1,2, … ,17 are the following (Knott, The
Fibonacci numbers, 2005):
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597
The Fibonacci sequence can be represented by a recurrence relation ,
where seeds 0 and 1. The same sequence can also be extended to negative index .
The sequence with negative index …,8,5,3,2,1,1,0,1,1,2,3,5,8,… can be satisfied by
1 .
One of the interesting properties of Fibonacci numbers is that every third number in the
sequence is even and every element of the sequence is always a multiple of . for this
reason, Fibonacci sequence satisfies a stronger divisibility property described by (Su)
(Ribenboim, 2000) (Renault, 1996):
gcd , ,
This unremarkable recursive sequence has yet another interesting pattern. The sides of
the squares in the figure 1 correspond to the numbers in the Fibonacci sequence and squares
are arranged in the outwardly spiraling pattern. Each rectangle in the image has roughly the
same shape and ratio for their length and width. It is also very interesting to note that as the
sequence of rectangles develops outwards, the ratio of length to width on every step is the
Fibonacci numbers Page 4