19. (FRG 2) Denote by $a_{n}$ the greatest number that is not divisible by 3 and that divides $n$. Consider the sequence $s_{0}=0, s_{n}=a_{1}+a_{2}+\cdots+a_{n}$, $n \in \mathbb{N}$. Denote by $A(n)$ the number of all sums $s_{k}\left(0 \leq k \leq 3^{n}, k \in \mathbb{N}_{0}\right)$ that are divisible by 3 . Prove the formula
$$A(n)=3^{n-1}+2 \cdot 3^{(n / 2)-1} \cos (n \pi / 6), \quad n \in \mathbb{N}_{0} .$$