Exploring Fibonacci sequence
The original problem that Fibonacci investigated in 1202 was about how fast rabbits could reproduce in ideal circumstances.
In the case of rabbits, reproduction could start at the age of 1 month.
The ideal circumstance for the Fibonacci sequence assumes no deaths and each rabbit pair always produced a male and female rabbits.
The Wikipedia diagram below shows the relationship between Fibonacci numbers and the golden ratio. This relationship will be discussed in this write up.
An Excel spreadsheet can be used to generate a Fibonacci sequence by putting the “seed” or starting values in the first two rows and dragging down a formula with the relationship shown above.
Also of interests are the ratios of the neighboring numbers. See table below.
Some very interesting results can be found by taking the ratio of two adjacent Fibonacci numbers.
As larger and larger Fibonacci numbers are used, the ratio appears to converge to a limiting value.
The relationship between columns is shown below.
b. The attached spread sheet explores sequences where the stating values, seeds, are some arbitrary integers other than 1.
If f(0)=1 and f(1) = 3, then your sequence is a Lucas Sequence. All such sequences, however, have the same limit of the ratio of successive terms.
It does not matter if the sequence starts out negative or larger than 1.