If someone didn't light Channukah lights on the first night, does he/she then say the third blessing (Sh'hecheyunu) on the next night?
The Shulchan Aruch in Orach Chayim 676:1 says that you should recite it on the second night after the regular two b'rachos. The same goes for any subsequent night if it is the first time you are lighting this year.
But there are two exceptions to this rule:
The first is if he knew on the first night that he was not going to light his own candles, and he said the Shehecheyanu blessing (and also the שעשה נסים blessing) when he saw someone else’s candles (if he was able to), he does not repeat the blessing on subsequent nights, as it says in 676:3.
The second is if he relied on his wife’s lighting on the first night, in which case it is considered as if he himself did the lighting, and therefore he does not repeat the Shehecheyanu blessing on a subsequent night, as the Mishnah Berurah teaches in Se'if Katan 7.