SCHWEITZER 1981 Problem 1

We are given an infinite sequence of $1$ 's and $2$ 's with the following properties:

(1) The first element of the sequence is $1$ .

(2) There are no two consecutive $2$ 's or three consecutive $1$ 's.

(3) If we replace consecutive $1$ 's by a single $2$ , leave the single $1$ 's alone, and delete the original $2$ 's, then we recover the original sequence.

How many $2$ 's are there among the first $n$ elements of the sequence?

P. P. Palfy