A famous Halacha establishes that if a person neglected to count one of the days of the Omer, he continues counting on the subsequent nights without a Beracha. The Terumat Ha'deshen (Rabbi Yisrael Isserlin, 1390-1460) raises the question of whether this would apply also to a person who cannot remember whether or not he counted one day of the Omer. Must he assume that he missed a day and must therefore omit the Beracha henceforth, or may he assume that he did count, such that hey can continue counting as usual, with a Beracha?
The Terumat Ha'deshen rules that a person in this case may continue counting with a Beracha, due to the principle of "Sefek Sefeka," or a "double doubt." There is a famous debate between the Behag ("Ba'al Halachot Gedolot," an anonymous Halachic work from the Geonic period) and Tosefot concerning the nature of the Sefirat Ha'omer obligation. The Behag perceived Sefirat Ha'omer as a single Misva that extends over a forty-nine-day period. As such, if a person neglected to count on any one of the forty-nine days, he can no longer fulfill the Misva. Tosefot, however, viewed each day's counting as a separate Misva, which does not depend on any other day's counting. As such, if a person missed a day or several days of counting, this does not affect his obligation on subsequent nights, and he therefore continues to count as usual.
Thus, if a person cannot remember whether or not he counted on a certain day of the Omer, on the subsequent nights he faces a "double doubt." First, he may have in fact counted, in which case he may continue counting with a Beracha. But in addition, even if he missed a day's counting, perhaps Halacha follows the position of Tosefot, that a missed day has no effect on the subsequent days of the Omer. The principle of "Sefek Sefeka" establishes that one may act leniently in situations of a "double doubt," and thus the Terumat Ha'deshen rules that in such a case one may continue counting with a Beracha. This ruling is codified by the Shulhan Aruch (Orah Haim 489).
It must be emphasized that this applies only to a person who cannot remember whether or not he counted on a certain day. If a person is certain that he missed a day of counting, then he continues counting without a Beracha, in deference to the Behag's position.
Summary: A person who forgot to count a day of the Omer continues counting on subsequent nights without a Beracha. If, however, a person is uncertain whether or not he counted on a certain day, he may continue counting as usual, with a Beracha.