曼德博集合可以用复二次多项式来定义：

• fc(z)=z2+c{\displaystyle f_{c}(z)=z^{2}+c\,}

其中 c{\displaystyle c} 是一个复数參数。

从 z=0{\displaystyle z=0} 开始对 fc(z){\displaystyle f_{c}(z)} 进行迭代：

• zn+1=zn2+c,n=0,1,2,...{\displaystyle z_{n+1}=z_{n}^{2}+c,n=0,1,2,...}

• z0=0{\displaystyle z_{0}=0\,}

• z1=z02+c=c{\displaystyle z_{1}=z_{0}^{2}+c=c\,}

• z2=z12+c=c2+c{\displaystyle z_{2}=z_{1}^{2}+c=c^{2}+c\,}

• ........

Examples:

(1) c=  1:                        0, 1, 2, 5, 26, 677, 458330, ....

(2) c=  0:                        0, 0, 0, ...

(3) c= -3:                        0, -3, 6, 33, 1086, 1179393, ...

(4) c= -1:                        0, -1, 0, -1, 0, -1, 0, ...

(5) c=1/4:                      0, 0.25, 0.31.., 0.34.., 0.37.., 0.38.., ...

(6) c=1/2:                      0, 0.5, 0.75, 1.06.., 1.62.., 3.15.., ...

(7) c=-0.125-0.75i:         0, -0.125+0.75i, -0.672+0.563i, 0, -0.125+0.75i, ....

.......

(Source:  https://www.youtube.com/watch?v=4LQvjSf6SSw, https://www.bilibili.com/video/av37982028/)  

References: 

[1] 大自然的分形几何学 最新修订本 Mandelbrot, 1997 (Freely downloaded from https://icaredbd.com:449/verify/verify_reCAPTHA.php)

[2] https://www.youtube.com/watch?v=4LQvjSf6SSw, https://www.bilibili.com/video/av37982028/

[3] https://en.wikipedia.org/wiki/Mandelbrot_set:

Fibonacci sequence in the Mandelbrot set

It can be shown that the Fibonacci sequence is located within the Mandelbrot Set and that a relation exists between the main cardioid and the Farey Diagram. Upon mapping the main cardioid to a disk, one can notice that the amount of antennae that extends from the next largest Hyperbolic component, and that is located between the two previously selected components, follows suit with the Fibonacci sequence. The amount of antennae also correlates with the Farey Diagram and the denominator amounts within the corresponding fractional values, of which relate to the distance around the disk. Both portions of these fractional values themselves can be summed together after 13{\displaystyle {\frac {1}{3}}}to produce the location of the next Hyperbolic component within the sequence. Thus, the Fibonacci sequence of 1, 2, 3, 5, 8, 13, and 21 can be found within the Mandelbrot set.

Freshman's sum (https://en.wikipedia.org/wiki/Mediant_(mathematics)):

"freshman's dream" also sometimes refers to the theorem that says that for a prime number p, if x and y are members of a commutative ring of characteristic p, then (x + y)^p = x^p + y^p.  (https://en.wikipedia.org/wiki/Freshman%27s_dream)

Sophomore's dream:

(https://en.wikipedia.org/wiki/Sophomore%27s_dream)

[4] http://mathcubic.org/article/article/index/id/422.html  混沌与分形（二）：分形的奇迹——分数维的曲线 (https://mp.weixin.qq.com/s?__biz=MzI3MzE3OTI0Mw==&mid=2247492489&idx=1&sn=1b2c4b501c109a3a199a6514d9668bc6&chksm=eb25948ddc521d9be14ff22bcb836876ea0efa0ec197f6793596364763e4632e8ebd731657af&mpshare=1&scene=1&srcid=0121NibBM0R9bKKLajwWPwHp#rd)

[5] 分形 —— 故事之外: https://www.ee.cityu.edu.hk/~gchen/pdf/FractalsStories.pdf

[6] 自然的奥秘：混沌与分形: http://www.global-sci.org/v1/mc/issues/3/no3/freepdf/35s.pdf