Tetra-series

[Gottfried]

Have you tried computing or plotting \( AS(x^{1/x}) \) or \( AS(x)^{1/AS(x)} \) yet? I would do this but I don't have any code yet for AS(x), and I'm lazy.

Andrew Robbins

Yes, I fiddled with this a bit.

First, it is obvious, that in

Sb(x) = x - b^x + b^b^x - ... + ...

there is some "self-similarity", if x itself is a powertower of base b

Sb(0) = s0
Sb(1) = -s0
Sb(b) = 1 - s0
Sb(b^b) = b - 1 + s0
and so on.

For the fixpoints, let t0,t1,t2,... be the fixpoints for base b
t_k = h_k(b)

then
Sb(t) = t - t + t - t +... -... = t*(eta(0)) = t/2

I've added some graphs for different bases for varying x, marking x=t0,t1 and x=0, x=1, x=b

It would be interesting to show graphs of negative heights, aka iterated logarithms...

Gottfried