Tahanun is a penitential supplication that makes up part of the daily prayer and, because of its solemn nature, it is omitted on joyous days (the holidays, in the presence of a groom, etc.). The prevailing Moroccan custom is that Tahanun is not recited for the first 12 days of the month of Sivan for the following reasons: The first day is Rosh Hodesh, on which Tahanun is never recited. On the second day of Sivan, Hashem commanded Bnei Israel to sanctify themselves before the Revelation at Mount Sinai. The third, fourth and fifth days commemorate the three days on which Bnei Israel prepared themselves to receive the Torah. The sixth day of Sivan is Shavuot, and Tahanun is not recited on the holidays. Tahanun is omitted on the next six days to commemorate the six days after Shavuot during which one still had the opportunity to offer the Hagiga sacrifice if unable to do so during the holiday itself.
Rabbi Itzhak ben Oualid (Vayomer Itzhak, Tefila 5) was of the opinion that Tahanun should be omitted on the 13th of Sivan as well, and cites an opinion to omit it even on the 14th. The 13th of Sivan can be explained by the fact that in the Diaspora, Shavuot falls on both the 6th and 7th of Sivan and so the six days commemorating the Hagiga would extend to the 13th of Sivan. The opinion regarding the 14th of Sivan is supported by the Kabbalah, which says that Tahanun is recited on 222 days of the year and it works out mathematically that Tahanun would not occur on the 14th. This mathematical calculation has been shown to be problematic and indeed it is not a widespread custom to omit Tahanun on the 14th of Sivan.
Summary: Unless one has a specific custom to omit Tahanun on the 13th and/or the 14th of Sivan, the prevailing Moroccan custom is that Tahanun is not recited for the first 12 days only.