This question is motivated by recent work of R P Stanley, Theorems and conjectures on some rational generating functions. Consider the polynomials $$P_n(x)=\prod_{i=1}^{n-1}(1+x^{3^{i-1}}+x^{3^i}).$$ Define the sequence $a_n$ to count the number of monomials of $P_n(x)$. For example, \begin{align*} P_2(x)&=x^3 + x + 1 \qquad \qquad \qquad \qquad \qquad \qquad\qquad \,\,\implies \qquad a_2=3, \\ P_3(x)&=x^{12} + x^{10} + x^9 + x^6 + x^4 + 2x^3 + x + 1 \qquad \implies \qquad a_3=8. \end{align*} Recall the Fibonacci numbers $F_1=F_2=1$ and $F_{n+2}=F_{n+1}+F_n$.

QUESTION.Is it true that $a_n=F_{2n}$? How does "ternary expansion" relate to Fibonacci?