Interesting Numbers

The Fibonacci sequence is one of the most widely used sequences when introducing the concept of sequences. The sequence is defined as the following:
F_N = F_{N – 1} + F_{N – 2}
So, starting with F₀ = 0 and F₁ = 1, the first few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, … and so on.

An interesting thing to note about the Fibonacci sequence is that the ratio of one Fib number to its predecessor approaches the Golden Ratio as the Fib numbers get higher and higher. For example:

Golden Ratio =  1.61803398875

2/1 = 2

3/2 = 1.5

5/3 = 1.667

8/5 = 1.6

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55/34 = 1.617647059
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987/610 = 1.618032787

This is a very interesting point to me. However, I learned a few days ago that there is another sequence like the Fibonacci sequence. If you take any two positive integers as the starting of the sequence, and the apply the Fibonacci method, the same approach to the Golden Ratio occurs, with a twist. Let’s test this:

Take the first number = 1234
Take the second number = 1588


1588/1234 = 1.286871961

1588+1234 = 2822

2822/1588 = 1.777078086

2822+1588 = 4410

4410/2822 = 1.562721474

from here, I will just show the divisions...

7232/4410 = 1.639909297

11642/7232 = 1.60978923

18874/11642 = 1.621199107

30516/18874 = 1.616827382

It seems as though the ratio is “hovered around” as the sequence grows. To cut down on the number of hand calculations, I wrote a python script which I have attached to this post. I took three samples and recorded the output. Sample1 I used A = 1234 and B = 1588, Sample 2 I used A = 10 and B = 20 and Sample 3 I used A = 13 and B = 17. I used L = 10000000000 for all three. They each settled down to approximately the Golden Ratio after about 20 iterations of the algorithm. I do not know the mathematical proof of this sequence however I find it interesting that they each reach the same result in a similar numbers of iterations. I’ve attached the python file and my three sample outputs if you’re interested in playing with the sequence! I’ve also linked some articles about this subject in case you want to read more technical descriptions.

More on this subject

Fibonacci Numbers
Golden Ratio

Downloads

interesting_sequence.py

Sample3 Sample2 Sample1
Thanks for reading!