A349710 Paschal full moon dates expressed as days after March 21 (Julian calendar).

15, 4, 23, 12, 1, 20, 9, 28, 17, 6, 25, 14, 3, 22, 11, 0, 19, 8, 27, 15, 4, 23, 12, 1, 20, 9, 28, 17, 6, 25, 14, 3, 22, 11, 0, 19, 8, 27, 15, 4, 23, 12, 1, 20, 9, 28, 17, 6, 25, 14, 3, 22, 11, 0, 19, 8, 27, 15, 4, 23, 12, 1, 20, 9, 28, 17, 6, 25, 14, 3, 22, 11, 0, 19, 8

Offset: 0

Author

Robert B Fowler, Jan 05 2022

Keywords

• easy
• nonn

Comments

The date of the Julian Paschal (Ecclesiastical) Full Moon (JPFM) in year n is a(n) days after March 21. Julian Easter Sunday is the first Sunday after (never on) the JPFM. The complete JPFM cycle is a repeating sequence of 19 terms, a(0) through a(18). The year 0 AD (conventionally denoted as 1 BC) is used as a starting point for a(0) solely as a computational convenience. The complete Julian Easter Sunday cycle is 19*4*7 = 532 years. For details on Easter and the Paschal Full Moon, in both Julian and Gregorian calendars, see A348924.

Examples

			For year 2021: n = 2021, m = 7, c = 20, q = 5, d = 13.
a(n) = 28 and s = 1, so the JPFM is April 18 and Julian Easter Sunday is April 19, which corresponds to May 2 in the Gregorian calendar.
		

References

• Byron Lawrence Gurnette and Richard van der Riet Woolley, Explanatory Supplement to the Astronomical Ephemeris, H. M. Stationery Office, London, 1961. Pages 420-422. The 1992 edition omits Julian Easter calculation.
• Edward Graham Richards, Mapping Time, Oxford University, London, 1998. Part IV, especially page 364.

Links

• Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Crossrefs

Cf. A348924.

Programs

Mathematica

a[n_] := Mod[19 * Mod[n, 19] + 15, 30]; Array[a, 100, 0] (* Amiram Eldar, Jan 05 2022 *)

Formula

n = calendar year (4 digits)

m = n mod 19 = position of n in the 19-year Metonic Lunar cycle

c = floor(n/100) = calendar century

q = floor(n/400) = calendar quad-century

d = c-q+2 = days to add to Julian calendar dates to convert to Gregorian

a(n) = days from March 21 to the JPFM (0 to 28 days)

= (19*m+15) mod 30

s = days from JPFM to next (Easter) Sunday (1 to 7 days)

= 7 - ((a(n)+floor(n*5/4)) mod 7)

Note that a(n) never equals 29, so Easter Sunday never falls on April 26.