Suppose we roll a fair $6$ sided die repeatedly. Find the expected number of rolls required to see $3$ of the same number in succession. 
Suppose we roll a fair six sided die repeatedly.
Find the expected number of rolls required to see $3$ of the same number in
succession

From the link below, I learned that $258$ rolls are expected to see 3 sixes appear in succession.  So I'm thinking that for a same (any) number, the rolls expected would be $258/6 = 43$. But I'm unsure how to show this and whether it really is correct.
How many times to roll a die before getting two consecutive sixes?

From Did's answer here, the probability generating function $u_0(s)=\mathbb{E}(s^T)$ for 
the number of trials $T$ needed to get three consecutive values the same is
$$u_0(s)={s^3\over 36-30s-5s^2}.$$ Differentiating this and setting $s=1$ in the 
derivative shows that $\mathbb{E}(T)=43.$
