I am wondering when, given $n \in (0,4]$, there exists a $\delta \in \mathbb{R}$ such that: $$i^n = \delta + \frac{1}{\delta + \frac{1}{\delta+\frac1{\ddots}}}$$ I thought I could just solve $i^n = \delta + \frac1{i^n}$, but this does not seem to give me all solutions. Example:
Let us find a general formula for $\delta$. If $i^n = \delta + \frac1{i^n}$, then we have $\delta = \frac{-i^n-1}{i^n}$. Using this to solve the $n=1$ case, we have $\delta = \frac{-i-1}{i}$, and this works, but $\delta$ is not an integer. But $\delta = 2$ is also a solution, which can be obtained by manipulating $i = \delta + \frac1i$.
So, I used two methods and found two different but viable solutions, and am left unsure whether I have found all solutions. However, I did find a $\delta \in \mathbb{Z}$ for $n=1$, so I am satisfied. The $n=2,3,4$ cases work out similarly. But I cannot find $\delta \in \mathbb{Z}$ for $n \notin \mathbb{N}$, but I also am not sure if I have found all solutions, and my example shows that the general formula does not always work.
So my question is: Does there always exist a $\delta \in \mathbb{Z}$ such that $i^n = \delta+\frac1{\delta+\frac1\ddots}$ for $\mathbb{n} \in (0,4]$? If not, what conditions must be placed on $n$ so that it is true?
If $n=1$ LHS is not real, and RHS is.