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A generalization of the Fibonacci sequence?

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Armachedes is not online. Armachedes
Joined: 30 May 2009
Total Posts: 3177
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.
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Armachedes is not online. Armachedes
Joined: 30 May 2009
Total Posts: 3177
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.
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