The following recurrence relation came up in some research I was working on:
$$x_n=\left(\frac{x_{n-1}}{n}\right)^2-a$$
Or equivalently the map:
$$z\mapsto\frac{z^2}{n^2}-a$$
Where $n$ is the iteration number. Specifically, I'm interested in the size of the convergence region across the real line. Some stuff I know about this map:
- For $a = 1$, it's easy, the "size on the real line" is $[-3,3]$.
I do have an infinite radical expansion for the size of the convergence region on the real line (see Solving the infinite radical $\sqrt{6+\sqrt{6+2\sqrt{6+3\sqrt{6+...}}}}$):
$$\sqrt{a+2\sqrt{a+3\sqrt{a+...}}}$$
That's why it's easy for $a=1$ -- it's just the Ramanujan radical, and equals 3. It's also easy for $a=0$ -- it's $\exp\left(-\mathrm{PolyLog}^{(1,0)}(0,1/2)\right)$ as per Wolfram Alpha.
Has anyone seen this map before? Here's the region of convergence on the complex plane, plotted numerically (for $a=6$):
Reminded me of this particular Julia set. It's the one generated from the Mandelbrot set's (0,0) point. Picture
Mandelbrot set on the left, Julia on the right.