Related for Rosh Hashanah 13:7

[The month] Nisan is the beginning of the year for the kings, pilgrimage festivals, months, and the collection of the temple tax. Some say: Also for renting houses. How [one should understand]: "for the kings"? [Answer:] If he dies in Adar and another takes his place in Adar, the year is counted both for him and for the other. If he dies in Nisan and another takes his place in Nisan, the year is counted both for him and for the other. If [one] dies in Adar and another takes his place in Nisan, the first [year] is counted for the first and the second [year] for the second.

All the congregational sacrifices are offered on the first of Nisan. And when the new moon comes in time, one sacrifice from the New, and if not, one sacrifice from the old. And if one offered [Pinge] for oneself, they were [as an offering] valid; only on the condition that they had to be handed over to the public.