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.

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

1,1

Comments

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.

Links

Table of n, a(n) for n=1..96.

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).

Formula

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).

Example

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.

Prog

(PARI) 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]}

Crossrefs

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

Sequence in context: A040347 A040346 A364786 * A124608 A218162 A064264

Adjacent sequences: A380151 A380152 A380153 * A380155 A380156 A380157

Keyword

nonn,fini

Author

Thomas Scheuerle, Jan 13 2025

Status

approved