The Fibonacci Sequence

The Fibonacci Sequence is by far the most famous and probably the oldest recursive sequence orginating from a problem published 1202 by Leonardo Pisano Fibonacci in his book Liber abacci.

It also represents the first and thus simplified attempt to describe dynamic structures within mathematics: A man puts a young pair of rabbits into a closed garden. After two months they start to produce offspring: two little rabbits every month. How many pairs of rabbits live in that garden after a year's time?

Counting the rabbits for, let's say seven months, we get the following table (k denotes a juvenile, k a one month old, K an adult rabbit):

Obviously the number of pairs coincides with the first seven Fibonacci numbers. From that we can solve the rabbit problem with the twelfth Fibonacci number: 144.

Besides this modest start of population dynamics the Fibonacci numbers unfold a surprising richness of mathematical coherence. There is for example a relation to the golden section

i

The golden section is the solution of a partition task: A given line of length $s>0$ is to be devided into two parts, so that the ratio 'whole line to bigger segment' is exactly the same as 'bigger segment to smaller one'. Let the given line be the interval $[0,s]$. Then we have to find a real number x from $]0,s[$ satisfying ${\displaystyle \frac{s}{x}=\frac{x}{s-x}\text{.}}$ Thus our problem is solved as soon as we could solve the quadratic equation $s^2-sx=x^2$ for x. With $x\in ]0,s[$ however we get from the p/q-formula: ${\displaystyle \begin{array}{ll}\hfill & x^2+sx-s^2=0\hfill \\ \iff \hfill & x=-\frac{s}{2}+\sqrt{\frac{s^2}{4}+s^2}\hfill \\ \iff \hfill & x=-\frac{s}{2}+\frac{s}{2}\sqrt{5}\hfill \\ \iff \hfill & x=s\cdot \frac{-1+\sqrt{5}}{2}\hfill \\ \iff \hfill & x=s\cdot \mathsf{\varphi }\text{.}\hfill \end{array}}$ So we get the value for x simply by multiplying the length s with the golden section number $\mathsf{\varphi }$.

which we will use to find a non-recursive representation for the Fibonacci sequence. To that end we need the golden section numbers

Consider:

There are some interesting properties of the section numbers. We need the following ones for the remainder.

• $\mathsf{\Phi }-\mathsf{\varphi }=1$

• $\mathsf{\Phi }\cdot \mathsf{\varphi }=1$

• $(x-\mathsf{\Phi })\cdot (x+\mathsf{\varphi })=x^2-(\mathsf{\Phi }-\mathsf{\varphi })x-\mathsf{\Phi }\cdot \mathsf{\varphi }=x^2-x-1$

  From the last equation we see that $\mathsf{\Phi }\text{ and }-\mathsf{\varphi }$ satisfy  $x^2-x-1=0$, that means:

  $\begin{array}{l}{\mathsf{\Phi }}^2=\mathsf{\Phi }+1\hfill \\ {(-\mathsf{\varphi })}^2=-\mathsf{\varphi }+1\text{.}\hfill \end{array}$
  
   

These preparations enable us to present a non-recursive version of the Fibonacci sequence.

There is much more information on Fibonacci numbers on the net. This site e.g. is very comprehensive.