21.2.1 * Find the number of natural numbers whose digits sum to $n$ and whose digits are 1 or 2.
Let the number of natural numbers that satisfy the given conditions be denoted as $a_{n}$, then $a_{1}=1, a_{2}=2$.
For natural numbers that satisfy the conditions and have a digit sum of $n+2$, if the unit digit is 1, there are $a_{n+1}$ such numbers; if the unit digit is 2, there are $a_{n}$ such numbers, thus $a_{n+2}=a_{n+1}+a_{n}$. By the method of characteristic roots, it is known that $a_{n}=$ $\frac{1}{\sqrt{5}}\left[\left(\frac{1+\sqrt{5}}{2}\right)^{n+1}-\left(\frac{1-\sqrt{5}}{2}\right)^{n+1}\right]$
Comment: This problem provides another combinatorial model for the Fibonacci sequence.