You can use any reasonable encoding. In the spirit of Godel, if you have a countable set of symbols you can represent a sequence of n them as a number by multiplying the product of the powers of the first n integers to the index of each symbol.
i.e. S_m1 S_m2 S_m3 ... S_mn is represented by 2^m1 * 3^m2 * 5^m3 * ... * p_n^mn
(m1, m2 ... mn are the integers giving the type of each of the symbols)
But then, I think you know that from reading Godel-related stuff. 
I am trying to understand Godel numbering, and any help would be much appreciated. For fun, I was just trying to make some Godel numbers for functions (is this even allowable?) to try to investigate properties of these functions with arithmetical proofs on the Godel numbers. So I had a few questions:
1. I can use induction on Godel numbered statements, right?
2. Is it fine to number functions?
3. If it is fine, how do I represent operations arithmetically. For example, how could I arithmetize something like g(f(x))?