Sunday, March 19, 2006
22:22
Okay, more about that Fibonacci variant. You've got an infinite sequence, right? It looks like this:
1, 1, 1, 2, 3, 4, 6, 9, 13, 19....
For any given number other than the first three, f(n) = f(n-1) + f(n-2). So I was wondering: Is it possible to tack numbers onto the left, so that the pattern continues to hold? Consider:
X, 1, 1, 1....
For the pattern to hold with the third 1, we need 1 + X = 1. Which gives X = 0. So now we have:
0, 1, 1, 1, 2, 3, 4, 6....
Tack on another one, and we get:
Y, 0, 1, 1...
So Y + 1 = 1, yielding Y = 0, and:
0, 0, 1, 1, 1, 2, 3....
Carry this process out a few more terms and you have:
....-1, 5, -3, -2, 3, 0, -2, 1, 1, -1, 0, 1, 0, 0, 1, 1, 1, 2, 3, 4, 6, 9, 13, 19....
Doesn't it seem weird that on the righthand side the sequence is monotonically increasing, probably with all positive derivatives, but on the lefthand side it's so chaotic?
The same sequence that's being generated off to the left can be formed in its own right with the following definition:
f(0)=f(1)=f(2)=1
f(n) = f(n-3) - f(n-2), n > 2
Thus for the first few terms we get:
1, 1, 1, 0, 0, 1, 0, -1, 1, 1, -2, 0, 3, -2, -3, 5, 1....
I've carried it out some distance using a calculator, and the numbers continue to fluctuate between positive and negative. The also seem to be generally increasing in absolute value, but it periodically dips back close to 0 as well, which means that it may have convergent subsequences as well as divergent ones in both directions.
1, 1, 1, 2, 3, 4, 6, 9, 13, 19....
For any given number other than the first three, f(n) = f(n-1) + f(n-2). So I was wondering: Is it possible to tack numbers onto the left, so that the pattern continues to hold? Consider:
X, 1, 1, 1....
For the pattern to hold with the third 1, we need 1 + X = 1. Which gives X = 0. So now we have:
0, 1, 1, 1, 2, 3, 4, 6....
Tack on another one, and we get:
Y, 0, 1, 1...
So Y + 1 = 1, yielding Y = 0, and:
0, 0, 1, 1, 1, 2, 3....
Carry this process out a few more terms and you have:
....-1, 5, -3, -2, 3, 0, -2, 1, 1, -1, 0, 1, 0, 0, 1, 1, 1, 2, 3, 4, 6, 9, 13, 19....
Doesn't it seem weird that on the righthand side the sequence is monotonically increasing, probably with all positive derivatives, but on the lefthand side it's so chaotic?
The same sequence that's being generated off to the left can be formed in its own right with the following definition:
f(0)=f(1)=f(2)=1
f(n) = f(n-3) - f(n-2), n > 2
Thus for the first few terms we get:
1, 1, 1, 0, 0, 1, 0, -1, 1, 1, -2, 0, 3, -2, -3, 5, 1....
I've carried it out some distance using a calculator, and the numbers continue to fluctuate between positive and negative. The also seem to be generally increasing in absolute value, but it periodically dips back close to 0 as well, which means that it may have convergent subsequences as well as divergent ones in both directions.
# posted by Fibonacci @ 19.3.06 
Comments:
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If you can find a group of rabbits that this models, please tell me about it. Keep in mind that you've got to account for the negative numbers.
You switched procedure when you went backwards:
Y, 0, 1, 1...
The ones in bold are the ones that should be combined to get 1. Where did "Y +1" come from?
It should be "Y + 0 = 1", yielding Y = 1
and 1, 0, 1, 1, 2, 3
Your sequence has several incongruities for the orignal.
0, 0, 1, 1, 1, 2, 3....
0 + 0 = 0 would yield an infinitely zero sequence in the traditional Fibonacci. The next two are okay. 0 + 1 = 1. But then 1 + 1 = 2, not 1. I think you threw an extra "1" in the original sequence.
Going back to the reconstructed sequence:
1, 0, 1, 1, 2, 3, we tack -1 on the left.
-1, 1, 0, 1, 1, 2, 3
2, -1, 1, 0, 1, 1, 2, 3
...-3, 2, -1, 1, 0, 1, 1, 2, 3...
Taking the left half, the seeds are 1, 1 if you want to continue the original Fibonacci backwards, with f(n) = f(n-1) - f(n-2)
1, 1, 0, 1, -1, 2, -3, 5, -8
Which is far less interesting than yours, just the orignal with alternating signs. ;^) But yours isn't the continuation of the original Fibonacci. I'm not sure where your extra beginning "1" came from or where you started skipping terms for the addition. *shrugs*
Y, 0, 1, 1...
The ones in bold are the ones that should be combined to get 1. Where did "Y +1" come from?
It should be "Y + 0 = 1", yielding Y = 1
and 1, 0, 1, 1, 2, 3
Your sequence has several incongruities for the orignal.
0, 0, 1, 1, 1, 2, 3....
0 + 0 = 0 would yield an infinitely zero sequence in the traditional Fibonacci. The next two are okay. 0 + 1 = 1. But then 1 + 1 = 2, not 1. I think you threw an extra "1" in the original sequence.
Going back to the reconstructed sequence:
1, 0, 1, 1, 2, 3, we tack -1 on the left.
-1, 1, 0, 1, 1, 2, 3
2, -1, 1, 0, 1, 1, 2, 3
...-3, 2, -1, 1, 0, 1, 1, 2, 3...
Taking the left half, the seeds are 1, 1 if you want to continue the original Fibonacci backwards, with f(n) = f(n-1) - f(n-2)
1, 1, 0, 1, -1, 2, -3, 5, -8
Which is far less interesting than yours, just the orignal with alternating signs. ;^) But yours isn't the continuation of the original Fibonacci. I'm not sure where your extra beginning "1" came from or where you started skipping terms for the addition. *shrugs*
*sighs* And suddenly I notice you were using a different variant of the original Fibonacci. Never mind.
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