Yes. A better way maybe just to notice what CaptainBlack said.
What I was getting at was that
And noting that on the RHS no two terms can be coalesced into a single term we can deduce that has as many terms as . But how many does that have?
Well it not only has the first term through the th term (which is terms) but it has the th term. So the number of terms in is . Consequently the number of terms in is