The Fibonacci sequence is one of the most widely used sequences when introducing the concept of sequences. The sequence is defined as the following:
FN = FN – 1 + FN – 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 . . . 55/34 = 1.617647059 . . . 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.