probability - Expected number of die rolls until obtaining three same consecutive numbers.

What is the expected number of die rolls required to get 3 same consecutive outcomes (for example: a 111, 222, etc) if we use a 6-sided fair die? I was able to solve the case for a particular number like 3 consecutive sixes. The answer comes out to be 258. But in this question can we say expected number is $1+E$, where $E$ is the expected number of obtaining two consecutive 1's if the first die roll was a one or two consecutive 2's if the first die will was a 2 etc.

So that way $E= \frac{5}{6}(1+E)+\frac{1}{6(\frac{5}{6(2+E)} + 2\frac{1}{6})}$. E comes out to be 42. So the final answer is 43. Is this correct? If not, what's the correct method?