Nested radicals and n-th roots

Question

There are many beautiful infinite radical equations, some relatively straightforward, some much more subtle: $$ x = \sqrt{ x \sqrt{ x \sqrt{ x \sqrt{ \cdots } } } } $$ $$ \sqrt{2} = \sqrt{ 2/2 + \sqrt{ 2/2^2 + \sqrt{ 2/2^4 + + \sqrt{ 2/2^8 + \sqrt{ \cdots}}}}} $$ $$ 3 = \sqrt{1 + 2\sqrt{1 + 3\sqrt{ 1 + 4\sqrt{ \cdots }}}} $$

But I have seen far fewer analogous equations for $n$-roots. Here is one: $$ 2 = \sqrt[3]{6 + \sqrt[3]{6 + \sqrt[3]{6 + \sqrt[3]{\cdots}}}} $$

My question is:

Q. Are there truly "more" beautiful infinite radical equations, or is it just our natural gravitation toward the simpler square-root equations that leads to collections emphasizing radicals?

I am aware this question is vague, but perhaps some nevertheless have insights.

Answer 1

For any integer $n \ge 2$ we have:

$n = \sqrt{(n^2-n)+\sqrt{(n^2-n)+\sqrt{(n^2-n)+\sqrt{(n^2-n)+\sqrt{(n^2-n)+\cdots}}}}}$.

This can be generalized for $m$-th roots:

$n = \sqrt[m]{(n^m-n)+\sqrt[m]{(n^m-n)+\sqrt[m]{(n^m-n)+\sqrt[m]{(n^m-n)+\sqrt[m]{(n^m-n)+\cdots}}}}}$.

This generates an infinite family of nested radical equations. Of course, this doesn't cover every nested radical equation out there. Now, how many equations in this infinite family are beautiful?

Answer 2

Here's a nested radical for 4th roots that is not so well known. Given the tetranacci numbers (an analogue of the fibonacci numbers). Let $y$ be the tetranacci constant, or the positive real root of,

$$y^4-y^3-y^2-y-1=0$$

Then,

$$y(3-y) = \sqrt[4]{41-11\sqrt[4]{41-11\sqrt[4]{41-11\sqrt[4]{41-\dots}}}} = 2.06719\dots$$

The family can be found in this MSE post.