Rolls until consecutive values differ by two

A fair six-sided die is rolled repeatedly. What is the expected number of rolls until two successive rolls differ by exactly $2$?

$17/3$. Group faces by how many follow-ups finish the game: faces $1,2,5,6$ have one partner and faces $3,4$ have two. Writing first-step equations for these two states, $E_1=1+\tfrac23E_1+\tfrac16E_2$ and $E_2=1+\tfrac13E_1+\tfrac13E_2$, gives $E_1=5$, $E_2=4$, and the total is $1+\tfrac23E_1+\tfrac13E_2=\tfrac{17}{3}$.