Is it possible to find an explicit function for any given recursive sequence?

Question

There are problems where we need to find an explicit formula for some recursive sequence such as the Fibonacci sequence, defined by $f_0=0, f_1=1,f_n=f_{n-1}+f_{n-2}$ where we can find $f_n$ in terms of $n$ only, or another example is $a_1=1,a_2=2,a_n=2a_{n-1}+3a_{n-2}$ where we can find $a_n$ in terms of $a_1,a_2,n$. I'm wondering if it is possible to find an explicit function for any given recursive sequence, in terms of at least one term of the set $\{a_1,a_2,...,a_k,n\}$.