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

%I A350458 #46 Jan 11 2025 13:47:59
%S A350458 347998,348353,348708,349091,349445,349800,350185,350539,350922,
%T A350458 351277,351631,352014,352369,352723,353108,353461,353815,354200,
%U A350458 354555,354938,355292,355647,356030,356385,356739,357124,357477,357861,358216,358571,358954,359308,359663
%N A350458 Chronological Julian day number of the first day (Tishri 1, Rosh Hashanah) of Hebrew calendar year n.
%C A350458 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).
%C A350458 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.
%C A350458 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.
%C A350458 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.
%C A350458 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.
%C A350458 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.
%D A350458 Louis A. Resnikoff, Jewish Calendar Calculations, Scripta Mathematica 9 (1943) 191-195, 274-277.
%D A350458 Edward Graham Richards, Mapping Time, Oxford University, London, 1998. Chapters 17 and 26.
%H A350458 E. G. Richards, <a href="/A057348/a057348.txt">Collected algorithms from Mapping Time</a>. The Calendar and Its History, Oxford University Press, 1998/1999.
%H A350458 P. K. Seidelman, Explanatory Supplement 1992, <a href="https://archive.org/details/131123ExplanatorySupplementAstronomicalAlmanac/page/n305/mode/2up">The Hebrew Calendar</a>.
%H A350458 Wikibooks, <a href="https://en.wikibooks.org/wiki/Mathematics_of_the_Jewish_Calendar">Mathematics of the Jewish Calendar</a>.
%H A350458 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hebrew_calendar">Hebrew calendar</a>.
%H A350458 Wikipedia, <a href="https://en.wikipedia.org/wiki/Hebrew_calendar#Months">Hebrew calendar - New Year months</a>.
%H A350458 Wikipedia, <a href="https://en.wikipedia.org/wiki/Julian_day">Julian day</a>
%H A350458 <a href="/index/Ca#calendar">Index entries for sequences related to calendars</a>
%F A350458 m(n) = floor((n*235 - 234)/19) = number of lunations since 0001-Tishri-1
%F A350458 j(n) = 347998 + floor((765433*m(n) + 12084)/25920) = JDN of lunation #m(n)
%F A350458 k(n) = j(n) + (floor(j(n)*6/7) mod 2)       (delay to avoid Wed, Fri, Sun)
%F A350458 a(n) = k(n) + 2    if k(n+1) - k(n) = 356   (delay to avoid 356-day year)
%F A350458      = k(n) + 1    if k(n) - k(n-1) = 382   (delay to avoid 382-day year)
%F A350458      = k(n)        otherwise
%F A350458 The delays to avoid 356-day and 382-day years occur about once in every 30.2 and 185.7 years, respectively.
%e A350458 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).
%e A350458 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).
%t A350458 m[n_] := Floor[(n*235 - 234)/19];
%t A350458 j[n_] := 347998 + Floor[(765433*m[n] + 12084)/25920];
%t A350458 k[n_] := j[n] + Mod[Floor[j[n]*6/7], 2];
%t A350458 a[n_] := If[k[n+1] - (kn = k[n]) == 356, kn+2, If[kn - k[n-1] == 382, kn+1, kn]];
%t A350458 Array[a, 30] (* _Amiram Eldar_, Jan 01 2022 *)
%Y A350458 Cf. A057349, A057350, A118661, A118662, A272699, A350471.
%K A350458 easy,nonn
%O A350458 1,1
%A A350458 _Robert B Fowler_, Jan 01 2022