Rolls until back-to-back sixes

A fair die is rolled repeatedly. What is the expected number of rolls until two sixes occur on consecutive rolls?

$42$. For a run of $k$ sixes the standard recursion gives $\sum_{j=1}^k 6^j$; for $k=2$ that is $6+36=42$. Equivalently, wait 6 rolls for a six; then the next roll finishes with probability $\tfrac16$, and otherwise you start over, so $E=6+1+\tfrac56E$.