Rolls needed to match or beat the opening roll

A fair six-sided die is rolled once and the result is noted. The die is then rolled again and again, stopping as soon as a roll is greater than or equal to that first result. If $N$ counts the rolls made after the first one, what is $\mathbb{E}[N]$?

$49/20$. If the first roll is $k$, each later roll succeeds with probability $(7-k)/6$, so the wait is geometric with mean $6/(7-k)$. Averaging over $k$ gives $\tfrac16\sum_{k=1}^6 \tfrac{6}{7-k}=1+\tfrac12+\dots+\tfrac16=\tfrac{49}{20}$.