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 Rabbi Yochanan: The shehechyanu blessing is recited on Shemini Atzeres. 

 Summary 

 A Proof: “The Parim Eilim and Kvasim all constitute sets, if one of them is missing the entire category of Korban cannot be brought.

 Rabbi Yehuda: The Parim do not constitute a set since their number is not constant and diminishes each day.

 They said to him: The Eilim and Kvasim also diminish on Shemini Atzeres!

 Rabbi Yehuda: Shemini is a separate Yom Tov. Just as the seven days of Sukkos require Korban [according to a certain order], Shir [the shir on sukkos discussed helping the poor], Bracha [it seems it requires a shehechyanu], and Linah [anyone who brings a korban must sleep over in Yerushalayim that night], so too, Shemini Atzeres requires Korban, Shir, Bracha and Linah”

So it seems that Shemini Atzeres requires a Shehechyanu. 

 Response: No, it is referring to Birkas Hamazon and Tefillah.