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

%I A057349 #58 Sep 11 2024 22:54:46
%S A057349 3,6,8,11,14,17,19,22,25,27,30,33,36,38,41,44,46,49,52,55,57,60,63,65,
%T A057349 68,71,74,76,79,82,84,87,90,93,95,98,101,103,106,109,112,114,117,120,
%U A057349 122,125,128,131,133,136,139,141,144,147,150,152,155,158,160,163,166
%N A057349 Leap years in the Hebrew Calendar starting in year 1 (3761 BCE). The leap year has an extra month.
%C A057349 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).
%C A057349 Also numbers m such that (1 + 7*m) mod 19 < 7.
%C A057349 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
%D A057349 N. Dershowitz and E. M. Reingold, Calendrical Calculations, Cambridge University Press, 1997.
%H A057349 Gergely Földvári, <a href="/A057349/a057349_1.jpg">Illustration of the correlation between leap years of the Hebrew calendar and the musical major scale</a>
%H A057349 Robinson Meyer, <a href="https://www.theatlantic.com/science/archive/2019/04/why-dont-easter-and-passover-always-line/587572/">The Ancient Math That Sets the Date of Easter and Passover: Why don't the two holidays always coincide?</a>, The Atlantic, April 19, 2019.
%H A057349 <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,1,-1).
%F A057349 a(n) = floor((19*n + 5)/7).
%F A057349 a(n) = A083033(n) + n + 2. - _Ralf Stephan_, Feb 24 2004
%F A057349 a(n) = A083089(n+1) + n. - _Robert B Fowler_, Dec 07 2022
%F A057349 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
%t A057349 LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {3, 6, 8, 11, 14, 17, 19, 22}, 70] (* _Harvey P. Dale_, Jan 18 2015 *)
%t A057349 Floor[(19Range[70] + 5)/7] (* _Alonso del Arte_, Apr 21 2019 *)
%o A057349 (PARI) a(n)=(19*n+5)\7 \\ _Charles R Greathouse IV_, Dec 07 2011
%Y A057349 Cf. A008685, Hebrew month pattern A057350, A057347.
%Y A057349 Cf. A350458 (JDN of Tishri 1 each year starting with year 1).
%Y A057349 Cf. A083033 (Dorian musical scale), A083089 (Lydian musical scale).
%K A057349 nonn,easy
%O A057349 1,1
%A A057349 _Mitch Harris_