Hello,

I came across something like this during my degree and I know of a way to show that any positive integer can be written uniquely in the form

where and .

The fact that each integer can be written uniquely in such a way implies to me that no such closed form expression could exist, since a closed expression could be re-written in many different ways. I don't know if this helps at all, but I can try to reproduce the proof if you'd like.