On some combinations of multiple zeta-star values

We prove that the sum of multiple zeta-star values over all indices inserted two 2’s into the string $(\underset{2n}{\underbrace{3,1,\mathrm{.}\mathrm{.}\mathrm{.},3,1}})$ is evaluated to a rational multiple of powers of ${\pi }^2$. We also establish certain conjectures on evaluations of multiple zeta-star values observed by numerical experiments.