# Iterative function with $z_{n+2}$

I'm currently playing arround with my custom fractal renderer and on Math SE in this answer Américo suggested the following function:

$z_{n+2}=z_{n+1}^{3}+c^{z_{n}}$

But to get the first value I'd need $z_{n+1}$, but for this I'd need $z_{n-1}$ and so on. How can I iterate this function?

• You can set $z_0$ and $z_1$ to some values and compute the rest of $z_n$ from there. Dec 4, 2014 at 10:38
• Thank you. Add it as an aswer and I'll mark it as correct :)
– Sebb
Dec 4, 2014 at 10:40

$$z_{n+1}=z_n^2+c$$
In Mandelbrot's sequence, $z_0$ is defined to be $0$. The definition of $z_{n+1}$ requires only $z_n$ to be defined, so a single starting value, $z_0$, is enough to define $z_n$ for all $n$.
$$z_{n+2}=z_{n+1}^{3}+c^{z_{n}}$$
In this sequence, to define $z_{n+2}$, you need both $z_{n+1}$ and $z_n$ to be defined. So, to define $z_n$ for all $n$, you need two starting values: $z_0$ and $z_1$. They can be of any value, but changing them will change the rest of the sequence.