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Numbers of tropical years that give successively better approximations (in the sense of continued fractions) of an integral number of synodic (lunar) months.
a(n) is the denominator of the n-th convergent of the ratio of the average length of the tropical year (current value: 365.242190 ephemeris days) to the average length of the lunar month (current value: 29.530589 ephemeris days) = 12.3682663...
The length of the tropical year and of the lunar month are affected by secular trends which change the value of the ratio by an order 10^-4 per millenium. Thus cycles of length much larger than 1000 years do not actually exist and the sequence is finite. The existence of the last term given here (a(8) = 1021) is indeed questionable (but see below the comment about the year 2017 which shows that a(8) could be useful even if a cycle of this length does not properly exist).
Values up to a(6) = 19 have been known and used since antiquity to build lunisolar calendars: a(4) = 8 is the octaeteris cycle and a(6) = 19 is the famous cycle of Meton of great accuracy for a rather small number of years.
The corresponding numbers of lunar months are in A280324 and the differences between tropical years and lunar months are in A280325.
Special year-2017 comment: the first year of the Julian calendar was year 45 B.C. and was supposed to begin at a new moon. This was 2061 years before 2017 (as there is no year 0) and 2061 = 2*a(8) + a(6). Therefore, the 2017 configuration of the Earth-Sun-Moon system should be very close to what it was in the first year of the Julian calendar. This is confirmed by the fact that the new moon was on New Year's Day 2017 minus a shift of 3 days (thus actually on Dec 29 2016). The 3-day shift is due to the error inherent in the Julian calendar's reckoning of leap years, not fully corrected by the Gregorian reform, which removed 10 days to correct the error accumulated since the time of the Council of Nicaea (AD 325). Had 3 additional days been removed to correct the error accumulated since 45 B.C., New Year's Day 2017 would have been a new moon day. The next similar fit (new moon 3 days before New Year's Day) is at the start of the year 2036.
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