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Constructing various functions of n

I’ve been trying to construct various functions like \$f(n)=n!\$, \$g(n)=n^2!\$, \$h(n)\$ is such that: \$h(1) = 1\$ and for \$ n > 1\$ \$h(n) = \log n \$ is there any such thing? I think such a function is a fractal function of n.
Is there any such function? If there is one, is there a parameter which would determine the “life” of such function? Does such a function exist? Does any well known function could be approximated by such functions?

A:

There is a type of function known as fractal functions. Some examples are the Sierpinski triangle and Cantor Dust.
For example, let \$f(n)\$ be equal to \$f^n(0)\$. Then:

\$f(0)=0\$
\$f(1)=1\$
\$f(2)=2\$
\$f(n)=f^{n-1}(1) + f^{n-2}(2)+…+f^1(n)\$

Also, if you write \$f(n)=n!\$ you can construct a type of fractal known as a Prouhet-Thue-Morse sequence. These can be written as integers written in base \$2\$: \$S_n=S_0S_1…S_{n-1}\$ with \$S_i=0\$ or \$S_i=1\$.
There are two sequences, \$M_n\$ and \$P_n\$.

\$M_0=0, M_1=1, M_2=1, M_3=0, M_4=1, M_5=0, M_6=0, M_7=0, M_8=1,…\$
\$P_0=0, P_1=1, P_2=0, P_3=0, P_4=1, P_5=0, P_6=