|
| 25 Dec 2011 03:16 AM |
There's no other forum with people remotely as smart as the people here, so this is why this is here. Sorry if this is spam.
I was thinking about the gamma function, which is a generalization of the factorial function such that,
Gamma[n] = (n-1)!
And the only thing you had to do to make the Gamma function work like the factorial function was to make two things true:
Gamma[0] = 1 Gamma[n+1] = n * Gamma[n]
So I was wondering if there was a generalization of the Fibonacci sequence that would work the same. The three requirements would be that,
Fib[0] = 0 Fib[1] = 1 Fib[n+2] = Fib[n+1] + Fib[n]
There are a ton more properties related to the Fibonaccis, but as far as I can see, if you satisfy those three all other properties are matched as well.
Can you think of a function that would do this? It would probably involve calculus, and maybe look something like the Gamma function.
The best present I can offer for Christmas is food for thought. |
|
|
| Report Abuse |
|
|
| 25 Dec 2011 03:30 AM |
| Herp. You could just use Binet's formula. Binet's for even and odd numbers, together with cos(pi*x) (to make non-integer values valid), could make something happen. |
|
|
| Report Abuse |
|