The modular group and words in its two generators

Consider the group, freely generated by the element $U$ of order $3$ and the element $S$ of order $2$. One of the models of such a group is the full modular group ${\sf PSL}_{2}(\mathbb{Z})$. Motivated by our investigations on quasi-modular forms and the Minkowski question mark function, we are lead to the following natural question. Some words in the alphabet $\{U,S\}$ are equal to the unity; for example, $USU^3SU^2$ is such a word of length $8$, and $USU^3SUSU^3S^3U$ is such a word of length $15$. Given $n\in\mathbb{N}_{0}$. Find the number of words of length $n$ which are equal to the unity. This is the new entry A265434 into the Online Encyclopedia of Integer Sequences. We investigate the generating function of this sequence and prove that it is an algebraic function over $\mathbb{Q}(x)$ of degree $3$.