A random question for mathematicians regarding i and the Fibonacci sequence.

[robo37]

What is the Fibonacci root of i, or to put it another way, what is the equation of x when       
 
 \( \Large \quad \quad      {({1+\sqrt 5\over 2})^x-(-1)^x ( { 2\over 1+\sqrt 5})^x \over \sqrt 5}=n \)  

?

All I need is a formular and I am capable of just entering the eqution and i straight into Google calculator to get a result.

I'm interested as decimal numbers entered into the Fibonacci function manly get complex results which shows a link between the function and i, and suggests that the Fibonacci root of i might not be complex and might be of some importance.

I know this isn't really to do with Tetration, but I've asked this across the internet and have not have any replies and I'll be greatfull for any help.

Thanks.

[Catullus]

What is the Fibonacci root of i, or to put it another way, what is the equation of x when ((1+sqrt 5)/2)^x-((-1)^x/((1+sqrt 5)/2)^x)))/sqrt 5=n?

You did not rationalize your denominator.
The numerical solutions to Fibonacci(x)=i are
x ~ -.42709179162288762 - .22105741463869769 * I
and
x ~ 1.67226840550786450 + 3.2642638352534426 * I.
I do not know of any closed forms for those numbers.
For an inverse to the Fibonacci function, there are no result found in terms of standard mathematical functions.