Could I represent any infinite digits sequence using finite formula? For example, I have set of digits 0,1,2,3,4,5,6,7,8,9. I'm writing any sequence using any amounts of each digit. Then, I want to represent my sequence as a finite formula to send it to my friend. Could somebody finish it? 

You could not represent any sequence of digits with a finite formula. The cardinal of the set of infinite digits sequence is $2^{\aleph_0}\geq\aleph_1$ and the cardinal of finite formulas is $\aleph_0$. So there is way too many sequences to stick a different formula on each.
