A057348 -id:A057348 - cp's OEIS frontend

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.

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).

A350539 Chronological Julian day number of the first day (Muharram 1) of Tabular Islamic year n.

Original entry on oeis.org

1948440, 1948794, 1949149, 1949503, 1949857, 1950212, 1950566, 1950921, 1951275, 1951629, 1951984, 1952338, 1952692, 1953047, 1953401, 1953755, 1954110, 1954464, 1954819, 1955173, 1955527, 1955882, 1956236, 1956590, 1956945, 1957299, 1957654, 1958008, 1958362, 1958717, 1959071, 1959425

Offset: 1

Robert B Fowler, Jan 04 2022

The Islamic calendar is purely lunar. It starts on Friday 0001-Mulharram-1 AH (Anno Hegirae) = AD 662-Jul-16 (Julian calendar) = AD 622-Jul-19 (Gregorian proleptic) = JDN 1948440. Every 12 months is a lunar year containing either 354 days (regular) or 355 days (leap year). Odd-numbered months are 30 days, even-numbered months are 29 days, except month 12 is 30 days in leap years. Each 30-year cycle contains 19 regular years and 11 leap years. Thus, 1 cycle = 30 lunar years = 360 lunar months = 10631 days, and a(n+30*k) = a(n) + k*10631, for all k. Since 10631 is not a multiple of 7, the calendar repeats after 7 cycles = 210 lunar years.
In various locations, the Islamic new moon is chosen to be dated by either (a) the first sighting of the lunar crescent, (b) astronomical new moon tables, or (c) tabular methods. Only the tabular methods are described here. At least five methods exist, differing only in the distribution of leap years. a(n) are calculated here using the most common method (Fazari or West Islam), in which the leap years within each 30-year cycle (first year of cycle is 1, not 0) are years {2, 5, 7, 10, 13, 16, 18, 21, 24, 26, 29} = {floor((30*k-1-c) / 11), k = 1..11, c = 3}. Three other tabular methods correspond to other values of c, namely, c = 4 (Kushyar or East Islam), c = 0 (Ismaili), c = -2 (Habash). In a fifth method (Fattuh), the leap years are not spaced evenly enough to fit this algorithm.
In a minority of locations, an epoch date of Thursday AD 662-Jul-15 is used; this subtracts one day from each of the five calculation methods.
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 29.530588865 days; the Islamic month averages 10631/360 = 29.5305555555 days, and falls behind the synodic moon by 0.04120 days per century. The astronomical tropical year averages 365.242192 days; the Islamic lunar year averages 12*10631/360 = 354.366666 days, so there are an average of 103.07120 Islamic years per tropical century.
The astronomical new moon of July 622 occurred on July 14 at 05:30 UT = 08:10 Mecca Local Mean Time (MLMT), but the crescent moon was not visible in Mecca until sunset of the next day July 15 (~18:00 MLMT), the start of 0001-Mulharram-1 AH, which is equated with AD 622-Jul-16 (which began 6 hours later at 24:00 MLMT). - Robert B Fowler, Aug 31 2024

			a(1) = floor((1*10631+3)/30) + 1948086 = 1948440 (JDN).
Year 1 has a(2) - a(1) = 354 days (a regular year).
Year 1 began on weekday (a(1) mod 7) = 4 (Friday).
Year 2 has a(3) - a(2) = 355 days (a leap year).

Jean Meeus, Astronomical Algorithms, Willmann-Bell, Richmond, Virginia. Second edition, 1998, chapter 9, pages 73-76.
Edward M. Reingold and Nachum Dershowitz, Calendrical Calculations, Cambridge University, UK. 1st edition, 1997, chapter 6 and appendix B8. 4th edition, 2018. Chapter 7 and Appendix D7.
Edward Graham Richards, Mapping Time, Oxford University, London, 1998. Chapter 15, pages 231-235, 311, 323-324.
Paul Kenneth Seidelmann and Leroy Elsworth Doggett, Explanatory Supplement to the Astronomical Almanac, Mill Valley, 1992. Pages 589-591.

Robert Harry van Gent, Islamic Calendar Converter, Mathematical Institute, Utrecht University, December 2019. An excellent source. The "Origin" section clearly describes how the various leap year methods are constructed; the "Islamic calendar converter" section converts Gregorian/Julian dates simultaneously into eight Islamic dates (four leap year methods in two epochs); the "Other online converters" section indicates which of the eight possible Islamic dates are used by ten online calendar converters, most of which do not divulge their method.
Dieter Koch & Alois Treindl, Astrodienst Planetary Positions for AD 622 July (page 272).
E. G. Richards, The identification of days, Figure 22.1 in: Mapping Time: The Calendar and Its History.
P. K. Seidelmann, The Islamic Calendar, Explanatory Supplement 1992.
Wikipedia, Islamic calendar.
Wikipedia, Tabular Islamic calendar.
Index entries for sequences related to calendars
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,-1).

Cf. A057347, A057348, A097105.

a[n_] := Floor[(n*10631 + 3)/30 + 1948086];
Array[a, 32] (* Amiram Eldar, Jan 04 2022 *)
IslamicNewYear[n_] := Module[{},
    date := DateObject[{n, 1, 1, 12},
            CalendarType -> "Islamic",
            TimeZone -> "Europe/London"];
    jl := CalendarConvert[date, "Julian"];
    jd := JulianDate[jl];
    MixedFractionParts[jd][[1]]
]; Table[IslamicNewYear[n], {n, 1, 32}] (* Peter Luschny, Feb 13 2022 *)

a(n) = floor((n*10631+c)/30) + 1948086.
c = 3 is used here; for other calendar methods, see comments section.
The epoch date of July 16 is assumed; for epoch July 15, subtract one from a(n).
Number of days in Islamic year n = a(n+1) - a(n).
Day of week of first day in year n = (a(n) mod 7) = 0 (Monday) to 6 (Sunday).
Julian day number of general Islamic date y,m,d = floor((y*10631+c)/30) + floor(m*59/2) + d + 1948056. Note that this single equation defines the entire Tabular Islamic calendar (for the four tabular methods mentioned in the comments).

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.

cp's OEIS Frontend

A056998 Erroneous version of A057348.

Views

Author

Keywords

Crossrefs

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.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica

Formula

A350539 Chronological Julian day number of the first day (Muharram 1) of Tabular Islamic year n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Formula