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A319280 Numbers that are congruent to {0, 4, 7, 11} mod 12.

%I A319280 #27 Jun 27 2025 17:19:42
%S A319280 0,4,7,11,12,16,19,23,24,28,31,35,36,40,43,47,48,52,55,59,60,64,67,71,
%T A319280 72,76,79,83,84,88,91,95,96,100,103,107,108,112,115,119,120,124,127,
%U A319280 131,132,136,139,143,144,148,151,155,156,160,163,167,168,172,175,179
%N A319280 Numbers that are congruent to {0, 4, 7, 11} mod 12.
%C A319280 Key-numbers of the pitches of a major seventh chord on a standard chromatic keyboard, with root = 0.
%H A319280 Jianing Song, <a href="/A319280/b319280.txt">Table of n, a(n) for n = 1..10000</a>
%H A319280 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A319280 a(n) = a(n-4) + 12 for n > 4.
%F A319280 a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
%F A319280 G.f.: x^2*(4 + 3*x + 4*x^2 + x^3)/((1 + x)*(1 + x^2)*(1 - x)^2).
%F A319280 a(n) = (6*n - 4 + (-1)^n + sqrt(2)*cos(Pi*n/2 + Pi/4))/2.
%F A319280 E.g.f.: ((6*x - 3)*cosh(x) + (6*x - 5)*sinh(x) + sqrt(2)*cos(x + Pi/4) + 2)/2.
%F A319280 Sum_{n>=2} (-1)^n/a(n) = log(3)/8 + log(2+sqrt(3))/(2*sqrt(3)) - 5*sqrt(3)*Pi/72. - _Amiram Eldar_, Dec 30 2021
%t A319280 Select[Range[0, 200], MemberQ[{0, 4, 7, 11}, Mod[#, 12]]&]
%t A319280 LinearRecurrence[{1, 0, 0, 1, -1}, {0, 4, 7, 11, 12}, 100]
%o A319280 (Magma) [n : n in [0..150] | n mod 12 in [0, 4, 7, 11]];
%o A319280 (PARI) my(x='x+O('x^99)); concat(0, Vec(x^2*(4+3*x+4*x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2)))
%Y A319280 A guide for some sequences related to modes and chords:
%Y A319280 Modes:
%Y A319280 Lydian mode (F): A083089
%Y A319280 Ionian mode (C): A083026
%Y A319280 Mixolydian mode (G): A083120
%Y A319280 Dorian mode (D): A083033
%Y A319280 Aeolian mode (A): A060107 (raised seventh: A083028)
%Y A319280 Phrygian mode (E): A083034
%Y A319280 Locrian mode (B): A082977
%Y A319280 Third chords:
%Y A319280 Major chord (F,C,G): A083030
%Y A319280 Minor chord (D,A,E): A083031
%Y A319280 Diminished chord (B): A319451
%Y A319280 Seventh chords:
%Y A319280 Major seventh chord (F,C): this sequence
%Y A319280 Dominant seventh chord (G): A083032
%Y A319280 Minor seventh chord (D,A,E): A319279
%Y A319280 Half-diminished seventh chord (B): A319452
%K A319280 nonn,easy
%O A319280 1,2
%A A319280 _Jianing Song_, Sep 16 2018