Rolling until a two or back-to-back ones

A fair six-sided die is rolled repeatedly, stopping either at the first $2$ or as soon as two $1$s appear on consecutive rolls, whichever happens first. What is the expected number of rolls?

$21/4$. Let $\mu$ be the expected rolls from scratch and $\mu_1$ just after a $1$. Then $\mu_1=1+\tfrac23\mu$ (a $1$ or $2$ ends it, anything else resets) and $\mu=1+\tfrac16\mu_1+\tfrac23\mu$. Substituting gives $\tfrac29\mu=\tfrac76$, so $\mu=\tfrac{21}{4}$.