i^i^i^i^... Is there a pattern? I was messing around with $i$ and I (haha) noticed that certain progressions arise when I keep on raising $i$ to $i$ to $i$ and so forth. Though, I am not really quite sure what is going on (and I don't have time to explore further).
In other words, is there an interesting pattern in the sequence:
$i$ , $i^i$, $i^{\left(i^i\right)}$, $i^{\left(i^{\left(i^i\right)}\right)}$, etc. 

Actually the limit exists.
Define $a_0=i$, $a_{n+1}=i^{a_n}$, $\lim_{n\to\infty}a_n=\frac{W(-\ln(i))}{-\ln(i)}\approx0.4383+0.3606i$, where $W(z)$ is the Lambert W function, $\ln(z)$ is the principle branch of $\log(z)$.
More generally, for each $z\in\mathbb{C}$, we can define such sequence $a_n(z)$, the limit exists only if $\frac{W(-\ln(z))}{-\ln(z)}$ is defined and they are equal.
Also the proof isn't hard, just messing with the definitions.
Correct me if there is any mistakes, I am just retrospecting what I read in high school.
Reference:


*

*http://en.wikipedia.org/wiki/Lambert%27s_W_function

*http://en.wikipedia.org/wiki/Tetration
