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


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?

  • 1
    $\begingroup$ You can set $z_0$ and $z_1$ to some values and compute the rest of $z_n$ from there. $\endgroup$
    – Regret
    Dec 4, 2014 at 10:38
  • $\begingroup$ Thank you. Add it as an aswer and I'll mark it as correct :) $\endgroup$
    – Sebb
    Dec 4, 2014 at 10:40

1 Answer 1



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 $n$.


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.


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