I'm sure you are aware of the summation types of sequences with the A's. B being some kind of multiplier.
A_{n}=BA_{(n-1)}+A_{(n-2)}
I'm not a mathematics guy. So forgive the mistakes. I was trying to find a faster way to climb via terms. I found a method that uses summation notation. I can skip them by a multiplier. But this is the first one I found that works for all of these summation sequences.
If I were to rewrite the above statement to something that is representitive of what I am showing it, I think it would look like this...
A_{2n-1}=(B_{n-1}^{2}-2)A_{2n-1}-1
A_{2n+1}=B_{n}-A_{2n-1}
I'll put this in the form that I understand, pseudo code.
Code:
A = 1
B = 3
exp = 8
for ( i = 1; i < exp; i++) {
A = B*A-1
B = B^{2}-2
}
B=B-A
When the initial B is equal to 3 it is a Fibonacci Number. When it is a 6, it is a Pell Number. This works for every integer. Give it a try, write the program and google the numbers it spits out. They wont match exactly with the ones we have because they generally have different formulas and methods. But they will be consistant with the sequence you choose.