A348924 -id:A348924 - cp's OEIS frontend

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

Robert B Fowler, Jan 05 2022

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.

			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.

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.

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).

Cf. A348924.

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

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.

A224110 Easter occurrences on March 22, March 23, ..., April 25 during a 5,700,000-year Gregorian Easter cycle.

Original entry on oeis.org

27550, 54150, 81225, 110200, 133000, 165300, 186200, 192850, 189525, 189525, 192850, 186200, 192850, 186200, 192850, 189525, 189525, 192850, 186200, 192850, 186200, 192850, 189525, 189525, 192850, 186200, 192850, 197400, 220400, 189525, 162450, 137750, 106400, 82650, 42000

Offset: 1

Rick L. Shepherd, Mar 31 2013

For the Gregorian calendar, the most common Easter Sunday date is the 19th of April and the least common is the 22nd of March with frequencies 3.867% and 0.483%. Among these 35 possible Easter dates there are only 16 distinct frequencies as there are sets of 6, 7, and 9 dates with equal frequencies for each date within the set (the larger the set the larger the frequency for each individual date).

The 16 distinct a(n) frequencies share many common factors. All are multiples of 25. All are multiples of either 56, 57, or 58. Except for a(28) and a(35), all are multiples of 19 (but the sum a(28)+a(35) is a multiple of 19). - Robert B Fowler, Jul 28 2024

			In an entire Gregorian Easter cycle, Easter Sunday occurs on March 22 exactly a(1) = 27550 times, so the frequency of occurrence of Easter on March 22 over the entire cycle is calculated simply as 27550/5700000 = 0.0048333..., rounding to 0.483%.  (Of course, these frequencies can vary within more-limited time periods.)  The van Gent link displays these frequencies as percentages and contains similar information for the Julian calendar.
The van Gent link does not currently contain these frequencies. The 35 a(n) values have been verified using A348924 for years 1 AD through 5700000 AD. - _Robert B Fowler_, Jul 28 2024

R. H. van Gent, Frequency of Easter Sundays. This link currently does not contain the actual frequencies, nor any algorithms to derive them; see A348924 below. - _Robert B Fowler_, Jul 28 2024
Ch. Zeller, Problema duplex calendarii fundamentale, Bulletin de la S. M. F., tome 11 (1883), pp. 59-61.

Cf. A348924 (algorithms for Paschal Full Moon and Easter Sunday in both Julian and Gregorian Calendars). - Robert B Fowler, Jul 28 2024

Cf. A375027 (frequencies for Julian Easter dates). - Robert B Fowler, Jul 29 2024

See the FORMULA at A348924.

A375027 Number of occurrences of Easter Sunday on March 22, March 23, ..., April 25 during a 532-year Julian Easter cycle.

Original entry on oeis.org

4, 8, 8, 12, 16, 16, 20, 16, 16, 20, 16, 16, 20, 16, 20, 20, 16, 20, 16, 16, 20, 16, 16, 20, 16, 20, 16, 16, 20, 16, 12, 12, 8, 8, 4

Offset: 1

Robert B Fowler, Jul 28 2024

During any 532-year range of the Julian Calendar, each of the 35 possible dates for Easter Sunday occurs either 4, 8, 12, 16, or 20 times. This cycle is much simpler than the Gregorian Easter cycle (A224110).

Cf. A224110 (frequencies for Gregorian Easter Sunday dates).
Cf. A348924 (algorithms for Paschal Full Moon and Easter Sunday in both Julian and Gregorian Calendars).
Cf. A349710 (algorithms for Paschal Full Moon and Easter Sunday in Julian Calendar).

Use the Julian Easter algorithm in A348924 for years 1 to 532 (or any range of 532 years), and tally the occurrence of each Easter date between March 22 and April 25.

A380154 Golden numbers, for the years of the Metonic cycle. Assigned to the full moon days of the year with the standard pattern of a Runic calendar. Days without assignment are represented by zero.

Original entry on oeis.org

19, 8, 0, 16, 5, 0, 13, 2, 0, 10, 0, 18, 7, 0, 15, 4, 0, 12, 1, 0, 9, 0, 17, 6, 0, 14, 3, 0, 11, 19, 0, 8, 0, 16, 5, 0, 13, 2, 0, 10, 0, 18, 7, 0, 15, 4, 0, 21, 1, 0, 9, 0, 17, 6, 0, 14, 3, 0, 11, 19, 8, 0, 16, 5, 0, 13, 2, 0, 10, 0, 18, 7, 0, 15, 4, 0, 12, 1, 0, 9, 0, 17, 6, 0, 14, 3, 0, 11, 19, 0, 8, 0, 16, 5, 0, 13

Offset: 1

Thomas Scheuerle, Jan 13 2025

The 19 year Metonic cycle repeats the pattern: 19,*,8,,16,5,,13,2,,10,,18,7,,15,4,,12,1,,9,,17,6,,14,3,,11. The gap marked with * alternates between odd and even cycles.
The alignment of the pattern is based on MS2913 from 1636. This is not accurate anymore in modern days as this pattern needs a shift of one day approximately every 304 years (Hipparchic cycle).
To each year the numbers from 1-19 where cyclically assigned. The positions of these numbers on the days of the year show the full moon for the corresponding year. Beware there are also calendar designs which track new moons with a cycle of 19 numbers.
The numbers are represented either by assigning runes from an extension of the Younger Futhark with three additional runic symbols or by the usage of Pentadic numerals. The latter is the case in MS2913.

			Note: In the original calendar usage no mapping on month and leap years was considered, instead the start of the year was determined by measuring the solstice each year. Days where originally numbered in 52 "weeks" of 7 days, however the counting restarted with 1 at each start of the year.
.
January
------------------------------------------------------------------
Day            1| 2| 3| 4| 5| 6| 7| 8| 9|10|11|12|13|14
Golden number 19| 8| 0|16| 5| 0|13| 2| 0|10| 0|18| 7| 0
------------------------------------------------------------------
Day           15|16|17|18|19|20|21|22|23|24|25|26|27|28|29|30|31
Golden number 15| 4| 0|12| 1| 0| 9| 0|17| 6| 0|14| 3| 0|11|19| 0
.
February
------------------------------------------------------------------
Day            1| 2| 3| 4| 5| 6| 7| 8| 9|10|11|12|13|14
Golden number  8| 0|16| 5| 0|13| 2| 0|10| 0|18| 7| 0|15
------------------------------------------------------------------
Day           15|16|17|18|19|20|21|22|23|24|25|26|27|28
Golden number  4| 0|12| 1| 0| 9| 0|17| 6| 0|14| 3| 0|11
.
March
------------------------------------------------------------------
Day            1| 2| 3| 4| 5| 6| 7| 8| 9|10|11|12|13|14
Golden number 19| 8| 0|16| 5| 0|13| 2| 0|10| 0|18| 7| 0
------------------------------------------------------------------
Day           15|16|17|18|19|20|21|22|23|24|25|26|27|28|29|30|31
Golden number 15| 4| 0|12| 1| 0| 9| 0|17| 6| 0|14| 3| 0|11|19| 0
.
April
------------------------------------------------------------------
Day            1| 2| 3| 4| 5| 6| 7| 8| 9|10|11|12|13|14
Golden number  8| 0|16| 5| 0|13| 2| 0|10| 0|18| 7| 0|15
------------------------------------------------------------------
Day            15|16|17|18|19|20|21|22|23|24|25|26|27|28|29|30
Golden number   4| 0|12| 1| 0| 9| 0|17| 6| 0|14| 3| 0|11|19| 8
.
May
------------------------------------------------------------------
Day            1| 2| 3| 4| 5| 6| 7| 8| 9|10|11|12|13|14
Golden number  0|16| 5| 0|13| 2| 0|10| 0|18| 7| 0|15| 4
------------------------------------------------------------------
Day            15|16|17|18|19|20|21|22|23|24|25|26|27|28|29|30|31
Golden number   0|12| 1| 0| 9| 0|17| 6| 0|14| 3| 0|11|19| 0| 8| 0
.
June
------------------------------------------------------------------
Day            1| 2| 3| 4| 5| 6| 7| 8| 9|10|11|12|13|14
Golden number 16| 5| 0|13| 2| 0|10| 0|18| 7| 0|15| 4| 0
------------------------------------------------------------------
Day            15|16|17|18|19|20|21|22|23|24|25|26|27|28|29|30
Golden number  12| 1| 0| 9| 0|17| 6| 0|14| 3| 0|11|19| 8| 0|16
.
July
------------------------------------------------------------------
Day            1| 2| 3| 4| 5| 6| 7| 8| 9|10|11|12|13|14
Golden number  5| 0|13| 2| 0|10| 0|18| 7| 0|15| 4| 0|12
------------------------------------------------------------------
Day            15|16|17|18|19|20|21|22|23|24|25|26|27|28|29|30|31
Golden number   1| 0| 9| 0|17| 6| 0|14| 3| 0|11|19| 0| 8| 0|16| 5
.
August
------------------------------------------------------------------
Day            1| 2| 3| 4| 5| 6| 7| 8| 9|10|11|12|13|14
Golden number  0|13| 2| 0|10| 0|18| 7| 0|15| 4| 0|12| 1
------------------------------------------------------------------
Day            15|16|17|18|19|20|21|22|23|24|25|26|27|28|29|30|31
Golden number   0| 9| 0|17| 6| 0|14| 3| 0|11|19| 8| 0|16| 5| 0|13
.
September
------------------------------------------------------------------
Day            1| 2| 3| 4| 5| 6| 7| 8| 9|10|11|12|13|14
Golden number  2| 0|10| 0|18| 7| 0|15| 4| 0|12| 1| 0| 9
------------------------------------------------------------------
Day            15|16|17|18|19|20|21|22|23|24|25|26|27|28|29|30
Golden number   0|17| 6| 0|14| 3| 0|11|19| 0| 8| 0|16| 5| 0|13
.
October
------------------------------------------------------------------
Day            1| 2| 3| 4| 5| 6| 7| 8| 9|10|11|12|13|14
Golden number  2| 0|10| 0|18| 7| 0|15| 4| 0|12| 1| 0| 9
------------------------------------------------------------------
Day            15|16|17|18|19|20|21|22|23|24|25|26|27|28|29|30|31
Golden number   0|17| 6| 0|14| 3| 0|11|19| 8| 0|16| 5| 0|13| 2| 0
.
November
------------------------------------------------------------------
Day            1| 2| 3| 4| 5| 6| 7| 8| 9|10|11|12|13|14
Golden number 10| 0|18| 7| 0|15| 4| 0|12| 1| 0| 9| 0|17
------------------------------------------------------------------
Day            15|16|17|18|19|20|21|22|23|24|25|26|27|28|29|30
Golden number   6| 0|14| 3| 0|11|19| 0| 8| 0|16| 5| 0|13| 2| 0
.
December
------------------------------------------------------------------
Day            1| 2| 3| 4| 5| 6| 7| 8| 9|10|11|12|13|14
Golden number 10| 0|18| 7| 0|15| 4| 0|12| 1| 0| 9| 0|17
------------------------------------------------------------------
Day            15|16|17|18|19|20|21|22|23|24|25|26|27|28|29|30|31
Golden number   6| 0|14| 3| 0|11|19| 8| 0|16| 5| 0|13| 2| 0|10| 0
.
Full moons in 2025:
((2025-5) mod 19) + 1 = 7.
This gives us the dates:
  13. January, 12. February, 13. March,
  12. April, 11. May, 10. June,
  9. July, 8. August, 6. September,
  6. October, 4. November, 4. December.

The Schoyen Collection, MS2913 Norwegian girdle calendar.
Wikipedia, Runic calendar.
Index entries for sequences related to calendars
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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).

Cf. A057349, A098476, A131773, A348924, A349710.

a(n) = {my(m = (n-1)%59 + 1); if(m >1, m++); [19, 0, 8, 0, 16, 5, 0, 13, 2, 0, 10, 0, 18, 7, 0, 15, 4, 0, 12, 1, 0, 9, 0, 17, 6, 0, 14, 3, 0, 11][(m-1)%30+1]}

Golden number (full moon) = ((year-5) mod 19) + 1. Example: 2025 gives 7.
There is also an estimation of new moon possible:
  Golden number (new moon) = ((year-1) mod 19) + 1. Example: 2025 gives 11.
Note: The modern definition of the Golden numbers is (Year mod 19) + 1 for new moons, but we adapt here to the offset in MS2913.
a(n) = a(n-59).

cp's OEIS Frontend

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

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A224110 Easter occurrences on March 22, March 23, ..., April 25 during a 5,700,000-year Gregorian Easter cycle.

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A375027 Number of occurrences of Easter Sunday on March 22, March 23, ..., April 25 during a 532-year Julian Easter cycle.

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A380154 Golden numbers, for the years of the Metonic cycle. Assigned to the full moon days of the year with the standard pattern of a Runic calendar. Days without assignment are represented by zero.

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