deforming geometric series

Yesterday I asked if there could be such a sequence of real numbers \{a_{n}\}_{\infty} such that for all n\in\mathbb{N}, AGM(a_{n},a_{n+1}) = 1/2^{n}, and it turns out that the new sequence is just a linearly stretched version, and that had me wondering, in what cases are we aware of nonlinearly stretched geometric sequences:

The ordinary geometric sequence:
\sum_{n=1}^{\infty} \frac{1}{2^{n}} = 1

The exponential:
\sum_{n=1}^{\infty} \frac{1}{2^{n}n!} = \sqrt{e}-1

“Wahi” or “Upsilon” functions:
\sum_{n=1}^{\infty} \frac{1}{2^{n}n^{n}} = \int_{0}^{1} \frac{dx}{2\sqrt{x^{x}}} = \Upsilon_{0}(1/2) = ʍ(1/2)-1

The polylogarithm (in particular, the dilogarithm):
\sum_{n=1}^{\infty} \frac{1}{2^{n}n^{2}} = Li_2(1/2)

But there are lacunary monsters like the following:
\sum_{n=1}^{\infty} \frac{1}{2^{n}3^{2^{n}}}
That there are no present methods of evaluation for.

automorphism groups of newton’s method fractals

This is just a stub. When I see things like this or this, I keep wondering about the automorphism groups of Newton’s method fractals.
Just as Kleinian group limit sets are the result of applying a transformation over and over again, Newton’s method limit sets are the result of applying Newton’s method over and over again on the complex plane. In some ways, Newton’s method also involves a curious sort of Indra’s Pearls effect. For instance, in these youtube videos:

It is remarkable how the bulbs seem to have ‘reflection’ to all other bulbs contained within. Moreover, each node where a bulb joins another bulb has the same arrangement of colors (roots) around it. I suppose that there is some group ཞ for a particular function f(z) for which f(z) = f(z)

gaps in zeta zeroes (attractor)

Using Odlyzko’s data (last file) for the nontrivial roots of the Riemann zeta function \zeta(s), I constructed an attractor — origin is in upper right corner, x increases as you go across and y increases as you go down. (x values are current differences (between nontrivial roots), y values are next differences (between nontrivial roots))