There are several irrational numbers that can be represented with continued fraction such that a mathematical rule arises in this continued fraction. For example, the Euler number $e$ can be represented as follows:
$$e=[2,1,2,1,1,4,1,1,6,1,1,8,1,1,10,\ldots]= 2+ \cfrac{1}{1+ \cfrac{1}{2 + \cfrac 1 \ddots}}$$
It's very fascinating that in the case where the number sequence after the $,$ is chaotic, it does exists a well-ordered scheme when considering continued fractions. Every three steps in the continued fraction, the parameter increases by $2$ and elsewhere the parameter is $1$. Why it is so? Another popular mathematical constant $\pi$ however, does not have a regular structure in the continued fraction:
$$\pi = [3,7,15,1,292,1,1,1,\ldots].$$
But I can see many $1$-s in this series of parameters. May it be that $\pi$ has an irregular continued fraction, but not a very intensive irregularity? Why these structural differences in continued fractions arise between $\pi$ and $e$?
And what is with other mathematical constant like the Euler-Mascheroni-constant $\Gamma$; is there also a regularity in the continued fractions?
I would be happy for any comments and answers.