I think my last post was poorly organized and my "finding" was really simplistic; I was subtracting Fibonacci numbers to get Fibonacci numbers which isn't profound. But I did explain the video in purely mathematical terms, mainly that the ones digit of the Fibonacci numbers in base 12 repeats in a pattern of 23 numbers. This is my new, better organized finding. I'm sure somebody else has found this before. I initially thought this could be used as a definition for the Fibonacci sequence but what I have researched so far doesn't help much. I've discovered the patterns in the ones digit for the Fibonacci numbers in bases 2-24.
I used the following Python script to do my research:
---
import string
abc = string.digits + string.letters
def base(number, radix):
"""base(number, radix)
inverse function to int(str,radix) and long(str,radix)
"""
if not 2 <= radix <= 36:
raise ValueError, "radix must be in 2..36"
result = []
addon = result.append
if number <>
number = -number
addon('-')
elif number == 0:
addon('0')
_divmod, _abc = divmod, abc
while number:
number, rdigit = _divmod(number, radix)
addon(_abc[rdigit])
result.reverse()
return ''.join(result)
a,b=0,1
print a
for n in range():
print base(b,radix)
a, b = b, a+b
-----
I stripped the ones digit manually. Any help on how to do that using Python would be greatly appreciated.
----
In base 2 the pattern "0,1,1" repeats in the ones digit.
In base 3 the pattern "0, 1, 1, 2, 0, 2, 2, 1" repeats in the ones digit.
Base 4 "0, 1, 1, 2, 1"
Base 5 "0, 1, 1, 2, 3, 0, 3, 3, 1, 4, 0, 4, 4, 3, 2, 0, 2, 2, 4, 1"
Base 6 "0, 1, 1, 2, 3, 5, 2, 1, 3, 4, 1, 5, 0, 5, 5, 4, 3, 1, 4, 5, 3, 2, 5, 1"
Base 7 "0, 1, 1, 2, 3, 5, 1, 6, 0, 6, 6, 5, 4, 2, 6, 1"
Base 8 "0, 1, 1, 2, 3, 5, 0, 5, 5, 2, 7, 1"
Base 9 "0, 1, 1, 2, 3, 5, 8, 4, 3, 7, 1, 8, 0, 8, 8, 7, 6, 4, 1, 5, 6, 2, 8, 1"
Base 10 "0, 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"
Base 11 "0, 1, 2, 3, 5, 8, 2, A, 1"
Base 12 "0, 1, 1, 2, 3, 5, 8, 1, 9, A, 7, 5, 0, 5, 5, A, 3, 1, 4, 5, 9, 2, B, 1"
Base 13 "0, 1, 1, 2, 3, 5, 8, 0, 8, 8, 3, B, 1, C, 0, C, C, B, A, 8, 5, 0, 5, 5, A, 2, C, 1"
Base 14 "0, 1, 1, 2, 3, 5, 8, D, 7, 6, D, 5, 4, 9, D, 8, 7, 1, 8, 9, 3, C, 1, D, 0, D, D, C, B, 9, 6, 1, 7, 8, 1, 9, A, 5, 1, 6, 7, D, 6, 5, B, 2, D, 1"
Base 15 "0, 1, 1, 2, 3, 5, 8, D, 6, 4, A, E, 9, 8, 2, A, C, 7, 4, B, 0, B, B, 7, 3, A, D, 8, 6, E, 5, 4, 9, D, 7, 5, C, 2, E, 1"
Base 16 "0, 1, 1, 2, 3, 5, 8, D, 5, 2, 7, 9, 0, 9, 9, 2, B, D, 8, 5, D, 2, F, 1"
Base 17 "0, 1, 1, 2, 3, 5, 8, D, 4, 0, 4, 4, 8, C, 3, F, 1, G, 0, G, G, F, E, C, 9, 4, D, 0, D, D, 9, 5, E, 2, G, 1"
Base 18 "0, 1, 1, 2, 3, 5, 8, D, 3, G, 1, H, 0, H, H, G, F, D, A, 5, F, 2, H, 1"
Base 19 "0, 1, 1, 2, 3, 5, 8, D, 2, F, H, D, B, 5, G, 2, I, 1"
Base 20 "0, 1, 1, 2, 3, 5, 8, D, 1, E, F, 9, 4, D, H, A, 7, H, 4, 1, 5, 6, B, H, 8, 5, D, I, B, 9, 0, 9, 9, I, 7, 5, C, H, 9, 6, F, 1, G, H, D, A, 3, D, G, 9, 5, E, J, D, C, 5, H, 2, J, 1
Base 21 "0, 1, 1, 2, 3, 5, 8, d, 0, d, d, 5, I, 2, k, 1"
Base 22 "0, 1, 1, 2, 3, 5, 8, D 1, C B 1, C B 1, C D 3, G, J, D, A, 1, B, C, 1, D, E, 5, J, 2, 1, 1"
Base 23 "0, 1, 1, 2, 3, 5, 8, d, L, B, 9, K, 6, 3, 9, C, L, A, 8, I, 3, L, L, M, 0, M, M, L, K, I, F, A, 2, C, E, 3, H, K, E, B, 2, D, F, 5, K, 2, M, 1"
Base 24 "0, 1, 1, 2, 3, 5, 8, D, L, A, 7, H, 0, H, H, A, 3, D, G, 5, L, 2, N, 1"
I wish there was a way to distinguish between an O (letter o) and a 0 (zero) on blogger and in Python, a 1 (one) and a l (letter L). I will tweek the program a bit to use upper case letters only and use a symbol instead of O and then check all of my work, because there may be some mistakes.
At some later date I may go to base 64. I stopped at base 24 because I saw the L problem arise, and I'm to tired to change the alter the program and persist. To go to 64 I'll have to use & or something for lower case letter L.
I thought there was a correlation between the base and the quantity of numbers in the pattern, and while there may be, I was wrong in my initial idea that it was (2b-1). While that works for base 12 and base 2 (which were the only I had studied at that point), it fails to work with other bases (not to mention I counted Base 12 wront.. The more I think about it, the numbers seem too random to work.
(2,3)
(3,8)
(4,5)
(5,20)
(6, 24)
(7, 16)
(8, 12)
(9, 24)
(10, 60)
(11, 9)
(12, 24)
(13, 28)
(14, 48)
(15, 40)
(16, 24)
(17, 36)
(18, 24)
(19, 18)
(20, 60)
(21, 16)
(22, 33)
(23, 48)
Also, I find it interesting that all of the patterns end with 1.
0 comments:
Post a Comment