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Numbers that are congruent to {0, 3, 7, 10} mod 12.
5

%I #23 Sep 08 2022 08:46:23

%S 0,3,7,10,12,15,19,22,24,27,31,34,36,39,43,46,48,51,55,58,60,63,67,70,

%T 72,75,79,82,84,87,91,94,96,99,103,106,108,111,115,118,120,123,127,

%U 130,132,135,139,142,144,147,151,154,156,159,163,166,168,171,175,178

%N Numbers that are congruent to {0, 3, 7, 10} mod 12.

%C Key-numbers of the pitches of a minor seventh chord on a standard chromatic keyboard, with root = 0.

%C Apart from the offset the same as A013574. - _R. J. Mathar_, Sep 27 2018

%H Jianing Song, <a href="/A319279/b319279.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-1).

%F a(n) = a(n-4) + 12 for n > 4.

%F a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.

%F G.f.: x^2*(3 + x + 2*x^2)/((x^2 + 1)*(x - 1)^2).

%F a(n) = (6*n - 5 + sqrt(2)*cos(Pi*n/2 + Pi/4))/2.

%F E.g.f.: ((6x - 5)*e^x + sqrt(2)*cos(x + Pi/4) + 4)/2.

%t Select[Range[0, 200], MemberQ[{0, 3, 7, 10}, Mod[#, 12]]&]

%t LinearRecurrence[{1, 0, 0, 1, -1}, {0, 3, 7, 10, 12}, 100]

%o (Magma) [n : n in [0..150] | n mod 12 in [0, 3, 7, 10]]

%o (PARI) x='x+O('x^99); concat(0, Vec(x^2*(3+x+2*x^2)/((x^2+1)*(x-1)^2)))

%Y A guide for some sequences related to modes and chords:

%Y Modes:

%Y Lydian mode (F): A083089

%Y Ionian mode (C): A083026

%Y Mixolydian mode (G): A083120

%Y Dorian mode (D): A083033

%Y Aeolian mode (A): A060107 (raised seventh: A083028)

%Y Phrygian mode (E): A083034

%Y Locrian mode (B): A082977

%Y Third chords:

%Y Major chord (F,C,G): A083030

%Y Minor chord (D,A,E): A083031

%Y Diminished chord (B): A319451

%Y Seventh chords:

%Y Major seventh chord (F,C): A319280

%Y Dominant seventh chord (G): A083032

%Y Minor seventh chord (D,A,E): this sequence

%Y Half-diminished seventh chord (B): A319452

%K nonn,easy

%O 1,2

%A _Jianing Song_, Sep 16 2018