A057349 -id:A057349 - cp's OEIS frontend

A057366 a(n) = floor(7*n/19).

0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 14, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 21, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 28, 28

Offset: 0

Mitch Harris

The cyclic pattern (and numerator of the gf) is computed using Euclid's algorithm for GCD.

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, NY, 1994.

G. C. Greubel, Table of n, a(n) for n = 0..5000
N. Dershowitz and E. M. Reingold, Calendrical Calculations Web Site
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0, 0,0,1,-1).

Similar pattern in Hebrew leap years A057349. Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

[Floor(7*n/19): n in [0..50]]; // G. C. Greubel, Nov 03 2017

Table[Floor[7*n/19], {n,0,50}] (* G. C. Greubel, Nov 03 2017 *)

a(n)=7*n\19 \\ Charles R Greathouse IV, Sep 02 2015

a(n) = a(n-1) + a(n-19) - a(n-20).

G.f.: x^3*(x^2-x+1)*(x^14 + x^13 + x^12 - x^10 + x^8 + x^7 + x^6 + x + 1)/( (x^18 + x^17 + x^16 + x^15 + x^14 + x^13 + x^12 + x^11 + x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x-1)^2 ). [Corrected by R. J. Mathar, Feb 20 2011]

A057347 Leap years in the Islamic calendar starting year 1 AH (Anno Hegirae) = 622 CE (Common Era or AD). There are 11 leap years in a 30 year cycle.

Original entry on oeis.org

2, 5, 7, 10, 13, 16, 18, 21, 24, 26, 29, 32, 35, 37, 40, 43, 46, 48, 51, 54, 56, 59, 62, 65, 67, 70, 73, 76, 78, 81, 84, 86, 89, 92, 95, 97, 100, 103, 106, 108, 111, 114, 116, 119, 122, 125, 127, 130, 133, 136, 138, 141, 144, 146, 149, 152, 155, 157, 160, 163, 166

Offset: 1

Mitch Harris

An Islamic year approximates 12 lunar months with 354 11/30 days (or 30 years with 10631 days).
Also, numbers m such that ((14 + 11*m) mod 30) < 11.
Worldwide, five different Islamic leap-year sequences are currently in use; this sequence (called "Fazari") is the most common of the five. See A350539. - Robert B Fowler, Dec 07 2022

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.

Index entries for sequences related to calendars

Cf. A057349 (Hebrew calendar leap years).
Cf. A057348 (Islamic months lengths), A008685 (Gregorian months lengths),
Cf. A350539 (Islamic New Year JDN and 8 Islamic calendar variants).

Select[Range[100], LeapYearQ[{#}, CalendarType -> "Islamic"]&] (* Jean-François Alcover, Apr 26 2020 *)

a(n) = floor((30*n - 4)/11).

A057350 Days in months in the Hebrew calendar starting from Nisan 5760 (Spring 2000 CE).

Original entry on oeis.org

30, 29, 30, 29, 30, 29, 30, 30, 30, 29, 30, 30, 29, 30, 29, 30, 29, 30, 29, 30, 29, 29, 29, 30, 29, 30, 29, 30, 29, 30, 29, 30, 29, 30, 29, 30, 29, 30, 29, 30, 29, 30, 29, 30, 30, 30, 29, 30, 30, 29, 30, 29, 30, 29, 30, 29, 30, 30, 30, 29, 30, 29, 30, 29, 30, 29, 30, 29, 30, 29, 29, 29, 30, 30, 29

Offset: 0

Mitch Harris

The months are Nisan, Iyyar, Sivan, Tammuz, Av, Elul, Tishri, Heshvan, Kislev, Teveth, Shevat, Adar. A leap month of 30 days, added between Shevat and Adar, is called Adar I, where the final month is then called Adar II. Heshvan and Kislev (the 8th and 9th months) may each have 29 or 30 days, depending on the calendrical rules.
36.22% of all years are "regular"; 25.51% are "deficient", with 29 days in both Heshvan and Kislev; 38.27% are "complete", with 30 days in both months. Note that Hebrew year 5760 is a 13-month leap year. - Robert B Fowler, Mar 08 2022
See A350458 for other comments, references, links, formulas, programs.

			A regular year has the pattern 30, 29, 30, 29, 30, 29, 30, 29, 30, 29, 30, 29.

N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.

Wikipedia, Hebrew calendar Four gates, Types of Hebrew calendar years and their frequencies.

Cf. A057349 (Hebrew leap years).

Cf. A350458 (Chronological JDN of Tishri 1 in Hebrew calendar year (AM) n).

A350458 Chronological Julian day number of the first day (Tishri 1, Rosh Hashanah) of Hebrew calendar year n.

Original entry on oeis.org

347998, 348353, 348708, 349091, 349445, 349800, 350185, 350539, 350922, 351277, 351631, 352014, 352369, 352723, 353108, 353461, 353815, 354200, 354555, 354938, 355292, 355647, 356030, 356385, 356739, 357124, 357477, 357861, 358216, 358571, 358954, 359308, 359663

Offset: 1

Robert B Fowler, Jan 01 2022

The Hebrew calendar in its current form was established between the 9th and 12th centuries AD; hence, earlier (proleptic) Hebrew dates do not always match actual historical dates. The starting year/month/day is Monday 0001-Tishri-1 AM (Anno Mundi) = 3761-Oct-7 BC (Julian proleptic) = 3761-Sep-7 BC (Gregorian proleptic) = JDN 347998 (chronological Julian day number).
The combined cycle of Hebrew months, weeks and days repeats every 689472 years = 8527680 months = 35975351 weeks = 251827457 days. Thus, a(n + 689472*k) = a(n) + 251827457*k, for all k.
The number of days in year n is a(n+1) - a(n), which is always either 353, 354, 355, 383, 384 or 385; this number determines the number of months (12 or 13), and the number of days in each month (29 or 30). The day of week of Tishri 1 is a(n) mod 7, which is 0 for Monday and 6 for Sunday.
Note that as many as four different Hebrew months are observed as the Jewish New Year for various purposes, resulting in different numbers for the months, but the year number always changes on Tishri 1, and the number of days in each month are determined by the Tishri New Year.
The chronological Julian day number (JDN) is the number of days since 4713-Jan-1 BC (Julian proleptic calendar), e.g., 2000-Jan-1 (Gregorian) = JDN 2451545. As used by historians, chronologers, and calendarists, it is an integer and does not incorporate time or location. The astronomical JDN incorporates both time and location: it equals the chronological JDN at UT (Greenwich) noon, and includes time as a decimal fraction of a day, e.g., JDN 2451545.50 = 2000-Jan-1 24:00 UT.
As of AD 2000, the astronomical synodic month averages 29.5305888645 days; the Hebrew lunar month averages 29.5305941358 days, and drifts ahead of the synodic month by 0.00652 days per century. The astronomical tropical year averages 365.2421926377 days; the Hebrew year averages 235/19 Hebrew months = 365.2468222060 days, and drifts ahead of the tropical year by 0.4629 days per century.

			For Hebrew year n=3 (3759 BC), m(3) = 24, j(3) = 348707, k(3) = 348708, k(2) = 348353, k(3) - k(2) = 355, k(4) = 349091, k(4) - k(3) = 383, a(3) = k(3) = 348708. Year 3 AM had 383 days and began on weekday 3 (Thursday). JDN 348708 = 3759-Sep-17 BC (Julian proleptic).
For Hebrew year n=5782 (AD 2021), m(5782) = 71501, j(5782) = 2459465, k(5782) = 2459465, k(5781) = 2459112, k(5783) = 2459849, a(5782) = k(5782) = 2459465. Year 5782 AM has 384 days and began on weekday 1 (Tuesday). JDN 2459465 = 2021-Sep-7 AD (Gregorian).

Louis A. Resnikoff, Jewish Calendar Calculations, Scripta Mathematica 9 (1943) 191-195, 274-277.
Edward Graham Richards, Mapping Time, Oxford University, London, 1998. Chapters 17 and 26.

E. G. Richards, Collected algorithms from Mapping Time. The Calendar and Its History, Oxford University Press, 1998/1999.
P. K. Seidelman, Explanatory Supplement 1992, The Hebrew Calendar.
Wikibooks, Mathematics of the Jewish Calendar.
Wikipedia, Hebrew calendar.
Wikipedia, Hebrew calendar - New Year months.
Wikipedia, Julian day
Index entries for sequences related to calendars

Cf. A057349, A057350, A118661, A118662, A272699, A350471.

m[n_] := Floor[(n*235 - 234)/19];
j[n_] := 347998 + Floor[(765433*m[n] + 12084)/25920];
k[n_] := j[n] + Mod[Floor[j[n]*6/7], 2];
a[n_] := If[k[n+1] - (kn = k[n]) == 356, kn+2, If[kn - k[n-1] == 382, kn+1, kn]];
Array[a, 30] (* Amiram Eldar, Jan 01 2022 *)

m(n) = floor((n*235 - 234)/19) = number of lunations since 0001-Tishri-1
j(n) = 347998 + floor((765433*m(n) + 12084)/25920) = JDN of lunation #m(n)
k(n) = j(n) + (floor(j(n)*6/7) mod 2)       (delay to avoid Wed, Fri, Sun)
a(n) = k(n) + 2    if k(n+1) - k(n) = 356   (delay to avoid 356-day year)
     = k(n) + 1    if k(n) - k(n-1) = 382   (delay to avoid 382-day year)
     = k(n)        otherwise
The delays to avoid 356-day and 382-day years occur about once in every 30.2 and 185.7 years, respectively.

A272699 Day of the week of Rosh Hashanah in Hebrew year n.

Original entry on oeis.org

2, 7, 5, 3, 7, 5, 5, 2, 7, 5, 2, 7, 5, 2, 2, 5, 2, 2, 7, 5, 2, 7, 5, 3, 7, 7, 3, 2, 7, 5, 3, 7, 5, 5, 2, 5, 5, 2, 7, 5, 2, 2, 5, 3, 2, 7, 5, 2, 7, 7, 3, 7, 7, 5, 2, 7, 5, 3, 7, 5, 5, 2, 5, 5, 2, 2, 5, 2, 2, 7, 3, 2, 7, 5, 3, 7, 7, 3, 7, 7, 5, 2, 7, 5, 5, 2, 5, 5, 2

Offset: 1

Nathan Fox, May 04 2016

nonn

Rosh Hashanah, the new year of the Hebrew Calendar, is the first day of the month of Tishri, which typically falls around the September equinox.
The holiday of Rosh Hashanah lasts for two days; this sequence considers only the first day.
The mathematical rules for determining the Hebrew Calendar stipulate that Rosh Hashanah cannot fall on Sunday, Wednesday, or Friday. So, every number in this sequence is 2, 3, 5, or 7.
The Hebrew Calendar is periodic with a period of 689472 years, so this sequence is periodic with period 689472.

			Rosh Hashanah 5776 fell on Monday, Sep 14 2015, so a(5776)=2.

Nathan Fox, Table of n, a(n) for n = 1..10000
Wikibooks, Mathematics of the Jewish Calendar.
D. Zeilberger, Maple program to determine Hebrew dates; Local copy

Cf. A057349, A057350.

A261190 Leap years in Symmetry454 calendar, starting year AD 1.

Original entry on oeis.org

3, 9, 15, 20, 26, 31, 37, 43, 48, 54, 60, 65, 71, 77, 82, 88, 93, 99, 105, 110, 116, 122, 127, 133, 139, 144, 150, 155, 161, 167, 172, 178, 184, 189, 195, 201, 206, 212, 217, 223, 229, 234, 240, 246, 251, 257, 263, 268, 274, 279, 285, 291, 296, 302, 308, 313

Offset: 1

Felix Fröhlich, Aug 11 2015

nonn

Year n is a leap year if and only if ((52 * n + 146) % 293) < 52.

Felix Fröhlich, Table of n, a(n) for n = 1..10000
I. Bromberg, 52/293 Symmetry454 (start Monday) Leap Years (Table of initial 780 terms)
I. Bromberg, The Symmetry454 Calendar - Leap Rule
Wikipedia, Symmetry454

Cf. A057347, A057349, A189917.

PARI
```
is(n) = ((52*n+146)%293) < 52
```

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

A057366 a(n) = floor(7*n/19).

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A057347 Leap years in the Islamic calendar starting year 1 AH (Anno Hegirae) = 622 CE (Common Era or AD). There are 11 leap years in a 30 year cycle.

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A057350 Days in months in the Hebrew calendar starting from Nisan 5760 (Spring 2000 CE).

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A350458 Chronological Julian day number of the first day (Tishri 1, Rosh Hashanah) of Hebrew calendar year n.

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A272699 Day of the week of Rosh Hashanah in Hebrew year n.

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A261190 Leap years in Symmetry454 calendar, starting year AD 1.

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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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