The verse in Vayikra (23:16) says:
עד ממחרת השבת השביעית תספרו חמישים יום
even unto the morrow after the seventh week shall ye number fifty days (OJPS, emphasis added)
In my experience the omer is only counted for 49 days. Is that accurate? In light of the above verse, why is that so?
For a start, the words before the ones you mention are:
שֶׁבַע שַׁבָּתוֹת, תְּמִימֹת תִּהְיֶינָה
Seven complete weeks are only 49 days.
Secondly, we see other times in Chumash that a number means "until, but not including" that number. For example in Devarim (25:3):
אַרְבָּעִים יַכֶּנּוּ, לֹא יֹסִיף
We only strike him 39 lashes; all the up to - but not including - 40.
That's also the simple explanation for the 70 souls that came down to Egypt with Yaakov: Breishith (46:27)
כָּל-הַנֶּפֶשׁ לְבֵית-יַעֲקֹב הַבָּאָה מִצְרַיְמָה, שִׁבְעִים
If you count them you only get to 69; all the up to - but not including - 70. (And yes, there's a known Medrash that the 70th was Yocheved born as they came through the gates.)
We do count 50 days. On day 50 we say in Kiddush "This day of Shavuot" which essentially means "Today is 50 days" since by definition "Shavuot" is the 50th day of the Omer count (not unlike counting in some other language or using a slang word for a number). The Rokeach 294 writes:
צריך לספור מ"ט יום אבל ביום נ' אין צריך לספור שהרי נזכר בברכה ובתפלה
One must count 49 days but on day 50 one need not count since it's mentioned in the blessing[s] and prayer[s].
Moreover, we count 50 days since we mark the 50th day on our calendars, just like a Zav (וספר לו) or Zavah (וספרה לה) counts 7 days. Sefira doesn't have to mean counting orally. As the Netziv (Haamek Davar Vayikra 23:15) writes: סתם ספירה משמעו בלב the word Sefirah by default means in one's heart.
While it's true that we are careful to count 49 days orally, some Rishonim (see some opinions in Tur OC 489 though this isn't a full survey of that issue) think that this only (or primarily) applies to counting weeks (eg. on day 8 you'd just say "Today is one week and one day"). Arguably the 7 weeks (and perhaps their constituent days) are required to be counted orally, while the 50 days are only required to be calculated. Indeed the Netziv (ibid.) argues that [the Talmud Menachot 65-66 (cf. Rabbenu Gershon there) implies that] the requirement to count orally is derived from the repetition of the command to count weeks in Devarim 16 (and note there is no repetition of the command to count days). See a similar line of reasoning in this article.
עד ממחרת השבת השביעת תספרו חמשים יום
Ralbag (ad loc.) and Tosafos (Menachos 65 amud 2 s.v. "Kasuv", first explanation) explain this as:
until the day after the seventh week you should count — which is day fifty.
Thus, you count 49.
