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Rosh Hashanah that falls on Saturday night:
When [the second night of] Rosh Hashanah falls on Motzei Shabbos [i.e. Saturday night], then in the evening prayer of Shemoneh Esrei the paragraph of Vetodieinu/ותודיענו is added prior to the paragraph of ותתן לנו, after the paragraph of Ata Vechartanu.
If one forgot to say Vetodieinu in Shemoneh Esrei what is he to do?
Some rule once he has begun saying “Vatiten Lanu Hashem” he is not to go back to say Vetodieinu and is rather to continue with Shemoneh Esrei as usual. In such a case he must beware to say Baruch Hamavdil prior to doing any Melacha which is permitted on Yom Tov.
Others however rule that one who does go back to say it has upon whom to rely so long as he has not yet finished the blessing of Mikadeish Yisrael Vehazmanim.
Kerias Shema Sheal Hamita:
The Maggid of Mezritch recited the Kerias Shema Sheal Hamita on the first night of Rosh Hashanah with great concentration and in great length. The Rebbe Rashab once took four hours to recite Kerias Shema Sheal Hamita on this night.
 As the first night of R”H can never fall on Motzei Shabbos. [See chapter 228/1; Kaf Hachaim 599/1]
 The reason: This follows the order of every Holiday that coincides with Motzei Shabbos. Now although we mention celebrating the Holiday in this paragraph and R”H is not a festival and neither is it one of the three Regalim, nevertheless there is no issue with saying it. The reason for this is because one does not intend to say that today is a festival and is simply counting the gifts that Hashem has given us of the holiness of Shabbos, the Honor of Yom Tov and the celebration of the festivals etc. [Admur 599/1; Tur; Levush; Taz 599/1; M”B 599/2; Kaf Hachaim 599/2]
 Mateh Efraim 599/7; See also Kaf Hachaim 599/3
 His reasoning is because doing so would require the person to repeat Hashem’s name in the blessing of Vatiten Lanu.
 Sheivet Halevy 9/23
 Sefer Hasichos 1942 p. 2
 Likkutei Sippurim 201