Start with $i=\sqrt{-1}$.

This will be $a_1$.

$a_2$ will be $i^i$.

$a_3$ will be $i^{i^{i}}$.

$\vdots$

etc.

$$a_n=i\uparrow\uparrow n$$

And, amazingly, you can evaluate $\lim_{n\to\infty}a_n=\lim_{n\to\infty}i\uparrow\uparrow n=e^{-W(-\ln(i))}\approx0.4383+0.3606i$.

You can check this, it does indeed converge to this value.

In fact, I decided to make a graph of $a_n$ to show that it converges. (y axis is imaginary part, x axis is real part.)

And, to little astonishment, I quickly noticed that there is an apparent pattern to the graph.

Commonly, we define $x\uparrow\uparrow0=1$, which I have included in the graph.

So the pattern seems very obvious. It follows a curved path that converges onto the point that was given above.

And, if you connect the dots, starting with the first point (given on the left as the first point) and trace a nice line to the second, third, and so fourth numbers, you will find an interesting spiral. I thought that at first, this spiral was writable as an equation, but apparently, there are a few implications.

You will notice that the blue dots are way closer to the converging point and that the red and black dots are a little closer. So whatever equation you can come up with should account that $a_{3n}$ is closest to the number you are trying to converge to.

I want (so desperately) to see if anyone can come up with an equation that allows the computation of $a_{0.5}$ that satisfies $$i^{a_{0.5}}=a_{1.5}$$a well known identity you can find on the Wikipedia.

At first glance of the graph I went on to think that perhaps, just perhaps, I (or you) could find a formula that allows us to define $i\uparrow\uparrow 0.5$.

If you are familiar with De'Moivres formula, it is a formula that allows us to perform compute

$$\sqrt{i}$$

with relative ease. It was derived when De'Moivre noticed an interesting pattern to $(a+bi)^n$. He proceeded to write his formula concerning the distance from zero and the angle from the positive real axis.

So I must tell you that I wish for the same to occur with $i\uparrow\uparrow n$. Perhaps the answer lies in using a different coordinate system. Perhaps the answer lies in calculating the distance one of the points on one of the lines (black, red, or blue) is from the converging spot and the adding in the angle at which the next point changes.

My progress on determining such a formula has gone nowhere. The most I can say is that $a_n$ is probably not chaotic and does indeed converge in a way that is most certainly not random.