How are Fibonacci numbers expressed in nature
Fibonacci numbersFibonacci sequences: These are consequences (an)n real numbers, which are defined as follows:
Be a0, a1 real numbers.
Define inductively for n ≥ 2
Now be a0 = 0, a1 = 1.
Question: For example, how big is a40? or also a300 ?
- It is a40 = 102.334.155 = 3 x 5 x 7 x 11 x 41 x 2161
And a300= 222232244629420445529739893461909967206666939096499764990979600
For n ≥ 1 let
|v (n) =|
So this is a vector in R.2.
Here are the first 8 vectors v (1), ..., v (8):
They are all close to the straight line with an incline
(You can formulate this as follows: the given value is just lim an-1/ an)
and indeed they "commute" around this straight line - why?
We operate here with the matrix
The following calculation rules apply to the eigenvalues:
It is advisable to also consider the following vector:
So it holds for all n ≥ 0:
- v (0)+ is an eigenvector with eigenvalue φ+,
- v (0)- is an eigenvector with eigenvalue φ-,
If we only consider the first coordinate (the x-coordinate) of v (n), we get:
This is called the Binet's formula (he published it in 1842, but it was already known to Euler and D. Bernoulli).
- Here, for clarification, the vectors v (0), ..., v (4):
The two inclined straight lines are the natural lines of the matrix A.
For each of the vectors v (0), ..., v (4) it is shown how it is the sum of a vector in the eigenline g+ = R.b+ and a vector in the eigenline g- = R.b- writes:
Since one knows that v (n) is an integer vector, one can in this way v (n) exactly to calculate! We are only interested in the x-coordinate of v (n), because this is the i-th Fibonacci number an.
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