How can a recurrence relation count subsets containing no consecutive integers?

To determine the number of subsets of the set of the first $n$ positive integers ($X_n$) that contain no two consecutive integers, we define this count as $f_n$.

A subset of this type is formed in one of two mutually exclusive ways:
* It is a valid subset of $X_{n-1}$, which accounts for all valid subsets that do not include the integer $n$.
* It is formed by adjoining the integer $n$ to a valid subset of $X_{n-2}$, which ensures that $n-1$ is not included, thereby maintaining the condition of no consecutive integers.

This logic leads to the recurrence relation:
$$f_n=f_{n-1}+f_{n-2}$$

With the initial values $f_0=1$ (the null set) and $f_1=2$ (the null set and the set $\{1\}$), this relation generates the sequence $f_2=3$, $f_3=5$, $f_4=8$, and so on. This approach allows for the calculation of the total count for any $n$ using generating functions or iterative computation.1