A057347 -id:A057347 - cp's OEIS frontend

A057367 a(n) = floor(11*n/30).

0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 11, 12, 12, 12, 13, 13, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 22, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 26, 26, 26, 27, 27, 27, 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
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).
Index entries for sequences related to Beatty sequences

Similar pattern in Islamic leap years A057347. Floors of other ratios: A004526, A002264, A002265, A004523, A057353, A057354, A057355, A057356, A057357, A057358, A057359, A057360, A057361, A057362, A057363, A057364, A057365, A057366, A057367.

[Floor(11*n/30): n in [0..50]]; // G. C. Greubel, Nov 03 2017

A057367:=n->floor(11*n/30); seq(A057367(k), k=0..100); # Wesley Ivan Hurt, Oct 29 2013

Table[Floor[11n/30], {n,0,100}] (* Wesley Ivan Hurt, Oct 29 2013 *)

a(n)=11*n\30 \\ Charles R Greathouse IV, Sep 02 2015

a(n) = a(n-1) + a(n-30) - a(n-31).

G.f.: x^3*(1 + x^3 + x^6 + x^8 + x^11 + x^14 + x^17 + x^19 + x^22 + x^25 + x^27)/( (1+x)*(1+x+x^2)*(x^2-x+1)*(x^4+x^3+x^2+x+1)*(x^4-x^3+x^2-x+1)*(x^8 - x^7 + x^5 - x^4 + x^3 - x + 1)*(x^8+x^7-x^5-x^4-x^3+x+1)*(x-1)^2 ). [Corrected by R. J. Mathar, Feb 20 2011]

A057349 Leap years in the Hebrew Calendar starting in year 1 (3761 BCE). The leap year has an extra month.

Original entry on oeis.org

3, 6, 8, 11, 14, 17, 19, 22, 25, 27, 30, 33, 36, 38, 41, 44, 46, 49, 52, 55, 57, 60, 63, 65, 68, 71, 74, 76, 79, 82, 84, 87, 90, 93, 95, 98, 101, 103, 106, 109, 112, 114, 117, 120, 122, 125, 128, 131, 133, 136, 139, 141, 144, 147, 150, 152, 155, 158, 160, 163, 166

Offset: 1

Mitch Harris

A Hebrew year approximates a solar year with 12 and 7/19 lunar months (or 19 years with 235 months, the 19-year Metonic cycle).
Also numbers m such that (1 + 7*m) mod 19 < 7.
In equal musical temperament, when an octave is divided into twelve half steps (a half step involves two notes and a whole step involves three notes, giving a total of thirteen notes including the octave), whole (w) and half (h) step intervals of the major scale follow a pattern of 2w-1h-3w-1h. Assigning the integer 2 (notes) to the half-step and 3 (notes) to the whole-step intervals will result in the same sequence when applied to the major scale. - Gergely Földvári, Jul 28 2024

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

Gergely Földvári, Illustration of the correlation between leap years of the Hebrew calendar and the musical major scale
Robinson Meyer, The Ancient Math That Sets the Date of Easter and Passover: Why don't the two holidays always coincide?, The Atlantic, April 19, 2019.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A008685, Hebrew month pattern A057350, A057347.
Cf. A350458 (JDN of Tishri 1 each year starting with year 1).
Cf. A083033 (Dorian musical scale), A083089 (Lydian musical scale).

LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {3, 6, 8, 11, 14, 17, 19, 22}, 70] (* Harvey P. Dale, Jan 18 2015 *)
Floor[(19Range[70] + 5)/7] (* Alonso del Arte, Apr 21 2019 *)

a(n)=(19*n+5)\7 \\ Charles R Greathouse IV, Dec 07 2011

a(n) = floor((19*n + 5)/7).
a(n) = A083033(n) + n + 2. - Ralf Stephan, Feb 24 2004
a(n) = A083089(n+1) + n. - Robert B Fowler, Dec 07 2022
G.f.: x*(2*x^6 + 3*x^5 + 3*x^4 + 3*x^3 + 2*x^2 + 3*x + 3)/((x - 1)^2*(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)). - Colin Barker, Jul 02 2012

A057348 Days in months in the Islamic calendar starting from Muharram, 1 AH. The twelfth month has 30 days in a leap year.

Original entry on oeis.org

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

Offset: 1

Mitch Harris

The months' names are Muharram, Safar, Rabi I, Rabi II, Jumada I, Jumada II, Rajab, Shaban, Ramadan, Shawwal, Dhu al-Qada, Dhu al-Hijja. The year, 354 11/30 days long, is defined as 12 lunar months and so travels around the solar year.
Periodic with period 360; the average month length is 29 + 191/360 = 29.530555... days. This is about 0.000032425 days shorter than the synodic month. - Charles R Greathouse IV, Mar 28 2016
Worldwide, five different Islamic leap-year cycles are currently in use; the month lengths a(n) correspond to the most common leap-year sequence (called "Fazari"). See A350539. - Robert B Fowler, Dec 07 2022

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

E. G. Richards, Collected algorithms from Mapping Time. The Calendar and its History, Oxford University Press 1999, (with corrections).
Index entries for linear recurrences with constant coefficients, order 360.

Cf. A057347, A350539.

isIslamicLeapYear := year -> irem(11 * year + 14, 30) < 11:
LastDayOfIslamicMonth := (year, month) -> if irem(month, 2) = 1 or (month = 12 and isIslamicLeapYear(year)) then 30 else 29 fi:
seq(seq(LastDayOfIslamicMonth(year, month), month=1..12), year = 1..5);
# Peter Luschny, Jan 07 2022

If[Mod[n, 2]==1, 30, If[Mod[n, 12]!=0, 29, If[Mod[14+11(Floor[(n-1)/12]+1), 30]<11, 30, 29]]]

a(n)=if(n%2,30,n%12,29,(n/12*11+14)%30<11,30,29) \\ Charles R Greathouse IV, Mar 28 2016

From Robert B Fowler, Dec 07 2022: (Start)
y = floor((n-1)/12) + 1     (Islamic year number)
m = ((n-1) mod 12) + 1      (Islamic month number)
a(n) = 29 + (m mod 2) + floor(m/12)*(floor((11*(y+1)+3)/30) - floor((11*y+3)/30)). (End)

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).

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A057367 a(n) = floor(11*n/30).

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A057349 Leap years in the Hebrew Calendar starting in year 1 (3761 BCE). The leap year has an extra month.

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A057348 Days in months in the Islamic calendar starting from Muharram, 1 AH. The twelfth month has 30 days in a leap year.

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

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

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