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I came across an essay by a man who claimed to find a cyclical pattern in the last digits of the Fibonacci numbers, which he wildly speculated / hyperbolized into evidence of divine intelligence, however despite my skepticism I did find it very intriguing and verified some of his results for myself - here's the link:

https://docs.google.com/document/d/1mVW ... -w42eU/pub

However, his fallacy, like that of a lot of numerology, is assuming a Base 10 number system. Just because humans have ten fingers doesn't mean all sapient beings do; there are any infinite number of bases to count in, and each one would provide different cyclical patterns in the Fibonacci numbers. So, I created a program that went through the first 2000 number bases, starting with binary, and found the cyclical pattern of the last digit in each number, outputting it to a file. I was interested to learn that the length of the cycle varies wildly from one base to the next... the writer of the above document believed that his period 60 cycle was universal and perfect, but there are an infinite number of other cycles hidden in the same sequence of Fibonacci numbers waiting to be found - who knows what discoveries similar to his we could make by exploring them?

Thus, I shall post the first 20 cycles, and you can peruse it at your leisure. Every so often I'll post more if anyone's interested, or I can post the Python code for writing the program to generate them yourself. The data I will provide is formatted as follows: base of the cycle (binary, ternary, decimal, hexadecimal, etc), length of the cycle (how many numbers in the series you have to go through before the cycle of last digits begins again), sum of the numbers in the cycle (sum of the recurring individual final digits), and then the full cycle itself. Read the document I linked to above (also, I wonder who the person is who created it?), and then you'll have a better idea of what I'm talking about and what it all means - and tell me if you find any interesting patterns!

NOTE: I just looked it up and discovered - apparently the cycles are called Pisano periods, but I don't know much more than that about them.

base 2 len 3 sum 2
[1, 1, 0]

base 3 len 8 sum 9
[1, 1, 2, 0, 2, 2, 1, 0]

base 4 len 6 sum 8
[1, 1, 2, 3, 1, 0]

base 5 len 20 sum 40
[1, 1, 2, 3, 0, 3, 3, 1, 4, 0, 4, 4, 3, 2, 0, 2, 2, 4, 1, 0]

base 6 len 24 sum 66
[1, 1, 2, 3, 5, 2, 1, 3, 4, 1, 5, 0, 5, 5, 4, 3, 1, 4, 5, 3, 2, 5, 1, 0]

base 7 len 16 sum 49
[1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1, 0]

base 8 len 12 sum 32
[1, 1, 2, 3, 5, 0, 5, 5, 2, 7, 1, 0]

base 9 len 24 sum 99
[1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 0, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1, 0]

base 10 len 60 sum 280
[1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1, 0]

base 11 len 10 sum 33
[1, 1, 2, 3, 5, 8, 2, 10, 1, 0]

base 12 len 24 sum 108
[1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0]

base 13 len 28 sum 156
[1, 1, 2, 3, 5, 8, 0, 8, 8, 3, 11, 1, 12, 0, 12, 12, 11, 10, 8, 5, 0, 5, 5, 10, 2, 12, 1, 0]

base 14 len 48 sum 322
[1, 1, 2, 3, 5, 8, 13, 7, 6, 13, 5, 4, 9, 13, 8, 7, 1, 8, 9, 3, 12, 1, 13, 0, 13, 13, 12, 11, 9, 6, 1, 7, 8, 1, 9, 10, 5, 1, 6, 7, 13, 6, 5, 11, 2, 13, 1, 0]

base 15 len 40 sum 285
[1, 1, 2, 3, 5, 8, 13, 6, 4, 10, 14, 9, 8, 2, 10, 12, 7, 4, 11, 0, 11, 11, 7, 3, 10, 13, 8, 6, 14, 5, 4, 9, 13, 7, 5, 12, 2, 14, 1, 0]

base 16 len 24 sum 144
[1, 1, 2, 3, 5, 8, 13, 5, 2, 7, 9, 0, 9, 9, 2, 11, 13, 8, 5, 13, 2, 15, 1, 0]

base 17 len 36 sum 272
[1, 1, 2, 3, 5, 8, 13, 4, 0, 4, 4, 8, 12, 3, 15, 1, 16, 0, 16, 16, 15, 14, 12, 9, 4, 13, 0, 13, 13, 9, 5, 14, 2, 16, 1, 0]

base 18 len 24 sum 198
[1, 1, 2, 3, 5, 8, 13, 3, 16, 1, 17, 0, 17, 17, 16, 15, 13, 10, 5, 15, 2, 17, 1, 0]

base 19 len 18 sum 133
[1, 1, 2, 3, 5, 8, 13, 2, 15, 17, 13, 11, 5, 16, 2, 18, 1, 0]

base 20 len 60 sum 560
[1, 1, 2, 3, 5, 8, 13, 1, 14, 15, 9, 4, 13, 17, 10, 7, 17, 4, 1, 5, 6, 11, 17, 8, 5, 13, 18, 11, 9, 0, 9, 9, 18, 7, 5, 12, 17, 9, 6, 15, 1, 16, 17, 13, 10, 3, 13, 16, 9, 5, 14, 19, 13, 12, 5, 17, 2, 19, 1, 0]

^These are not Fibonacci sequences. Can you perhaps clarify exactly what operation/function is performed upon the base? Fn for n = 0 or 1 or n-1 or n+1 id the base function of Fibonacci integer sequences.

What makes Fibonacci sequences interesting to me is not what numeric patterns emerge, but that each sequence always moves ever-toward or ever-away from the golden ration, depending upon where you start. It is the friction (to use an analogy) by which all energy is manifest. The pine cone spirals out from itself. Same for all life, actually. As the sequence grows, it becomes more like the Golden Ratio. However, it is in the germ of the sequence that the imperfection is introduced, the 0, 1 or 1, 1. Exponential growth can only occur after the initial spark. Picture the sequence as two dancers (the first two integers)... one will always be slightly ahead of the other, creating the dance. If nature were to follow the golden ratio, there would be only one perfect dancer, and no dance.

It is, as I said, the final DIGITS of Fibonacci sequences in every base from binary to base 20. Not the sequences themselves.