cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

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

Views

Author

Mitch Harris

Keywords

Comments

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

References

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

Links

Crossrefs

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).

Programs

Formula

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

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

Views

Author

Robert B Fowler, Jan 01 2022

Keywords

Comments

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.

Examples

			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).

References

  • 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.

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Nathan Fox, May 04 2016

Keywords

Comments

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.

Examples

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

Links

Crossrefs

Cf. A057349, A057350.
Showing 1-3 of 3 results.

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