Rolls until neighbouring values are within one

You keep rolling a fair six-sided die. What is the expected number of rolls until two consecutive rolls differ by at most $1$ (equal values count)?

$377/115\approx3.28$. Let $E_i$ be the expected extra rolls after a roll of $i$; symmetry pairs faces $1\leftrightarrow6$, $2\leftrightarrow5$, $3\leftrightarrow4$, leaving three linear first-step equations. Solving them gives an average of $\tfrac13(E_1+E_2+E_3)=\tfrac{262}{115}$ extra rolls after the first, so the total is $1+\tfrac{262}{115}=\tfrac{377}{115}$.