Artaxerxes I

Artaxerxes I was the King of Persia during the time of Nehemiah and Ezra. During his reign, he halted the rebuilding of the Temple area that his predecessor Cyrus had allowed. Later, he allowed it to continue. During the seventh year of his reign, he allowed Ezra and a large number of Israelites to return to Jerusalem. Then in his twenty-first year he allowed Nehemiah to return to Jerusalem, and to rebuild the city walls.

Seventy Weeks Prophecy
Artaxerxes is associated with a prophecy found in the Book of Daniel. The angel Gabriel came to Daniel and told him that the Jews would have 70 7s to finish the transgression against God, to be done with sin, for reconciliation to be made, to bring in everlasting righteousness, to complete God’s revelation, and to consecrate the Holy of Holies. Then the angel said that after the decree to rebuild Jerusalem until the Anointed One would die there would be 7 7s and 62 7s. It was Artaxerxes who made the decree in 444 BC in his 21st year. The "7s" are said to mean weeks of years, according to biblical scholars. After doing the math, there would be 49 years and 434 years, which adds up to 483 years. Now this is lunar years, since the all dates in the Old Testament follow the lunar calendar. To convert any Old Testament date to a Gregorian date (our calendar is called the Gregorian calendar), follow the following equation: $${x\cdot360 \over 365.25}=y$$ (x=biblical date, y=Gregorian date) After applying this method to the Daniel prophecy, from the decree to the Crucifixion, there were 476 years and 3 weeks. Add 476 years to 444 BC and the sum is 33 AD. If you consider that there was a lunar eclipse (that lasted 2 hours and 50 minutes) on April 3rd of 33 AD, and that April 3rd was right before the passover (meaning this was the day Jesus died on the cross), then we can work backwards to find out exactly when Artaxerxes gave the decree. This date would be Friday, March  13th,  444 BC. This date is Artaxerxes' 21 st year.

Year of the King
However the Book of Nehemiah says that it was his 20th year. Nehemiah was using a Jewish calendar that rendered the month of Tishri as the first month, which is in the fall. The decree was given in Nisan, in the spring. To Nehemiah, it was still Artaxerxes' 20th year. According to our form of reckoning time, it would have been the 21st year.