(08/11/2022, 10:09 PM)bo198214 Wrote: OMG, its as easy as this:
\[\text{fib}_\text{alt}(t) = \frac{\Phi^t - \cos(\pi t)\left|\Psi\right|^t}{\Phi-\Psi}\]
To be honest im getting a bit irritated.
1) This identity is like hundreds of years old and mentioned a thousand times and even on wiki.
2) the fibonacci sequence clearly is not an iteration since we have 0,1,1,2,... the occurence of 1 twice makes it not an iteration.
3) that identity does not satisfy the recursion f(x+1) = f(x) + f(x-1). It is just a lame cos used for a dubious unmotivated interpolation.
4) the real issue is that one of the eigenvalues has a negative value.
that is analogue to solving for real functions that are half-iterates of exp(- z).
5) without realizing 1 till 4 and adressing it or looking at or defining differently the fibonacci sequence we will not get anywhere.
6) if we take the matrix representation for other similar recursions that have only positive eigenvalues the solution is simple. Then again there was no issue , problem or question solved in that case.
7) there are some nice definitions or recursions for -say- the even fibonacci numbers.
things like f(x+1) = f(x) + f(x) + f(x-1) + f(x-2) + f(x-3) + f(x-4) + ...
( continuum sum recursion ideas flashbacks * intensifies * )
nice ... and not nice ; we get positive eigenvalues then and the case returns to trivial.
8. if you want something a bit new :
you might want to consider iterations of a^z + b^z, that seems closest to fibonnacci and analytic continu iterations.
or you might wanna find recursions for such exp sums. Well my suggestion ...
basically i felt the whole debate was running or circles or stating the obvious and well known.
regards
tommy1729