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

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

Author

Mitch Harris

Keywords

• nonn
• easy

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

• 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).

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

Mathematica

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 *)

PARI

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

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