Given that we have a recursive function defined as
$\begin{matrix} f_{0}=0 \\ f_{n}=f_{n-1}+1 \end{matrix}$
We can come up with a related non-recursive closed form function defined as
$g(x)=x$
Where for every value of $n$
$f_{n}=g(n)$
Do all recursive functions defined with
$\begin{matrix} f_{0}=\text{an integer} \\ f_{n}=\text{an algebraic function(1)} \end{matrix}$
(1): where said function involves only algebraic operations, like, addition, subtraction, multiplication, and division, as well as fractional or rational exponents
have such a related non-recursive closed form function $g(x)$?
I'm not looking for how to find $g(x)$ if it exists, I`m just wondering if one always exists.