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A083031 Numbers that are congruent to {0, 3, 7} mod 12.

%I A083031 #23 Sep 08 2022 08:45:10
%S A083031 0,3,7,12,15,19,24,27,31,36,39,43,48,51,55,60,63,67,72,75,79,84,87,91,
%T A083031 96,99,103,108,111,115,120,123,127,132,135,139,144,147,151,156,159,
%U A083031 163,168,171,175,180,183,187,192,195,199,204,207,211,216,219
%N A083031 Numbers that are congruent to {0, 3, 7} mod 12.
%C A083031 Key-numbers of the pitches of a minor common chord on a standard chromatic keyboard, with root = 0.
%H A083031 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F A083031 G.f.: x^2*(3 + 4*x + 5*x^2)/((1 + x + x^2)*(1 - x)^2). - _R. J. Mathar_, Oct 08 2011
%F A083031 From _Wesley Ivan Hurt_, Jun 14 2016: (Start)
%F A083031 a(n) = a(n-1) + a(n-3) - a(n-4) for n > 4.
%F A083031 a(n) = (12*n - 14 - cos(2*n*Pi/3) + sqrt(3)*sin(2*n*Pi/3))/3.
%F A083031 a(3k) = 12k - 5, a(3k-1) = 12k - 9, a(3k-2) = 12k - 12. (End)
%F A083031 a(n) = a(n-3) + 12 for n > 3. - _Jianing Song_, Sep 22 2018
%p A083031 A083031:=n->(12*n-14-cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/3: seq(A083031(n), n=1..100); # _Wesley Ivan Hurt_, Jun 14 2016
%t A083031 Select[Range[0, 400], MemberQ[{0, 3, 7}, Mod[#, 12]] &] (* _Wesley Ivan Hurt_, Jun 14 2016 *)
%t A083031 LinearRecurrence[{1, 0, 1, -1}, {0, 3, 7, 12}, 100] (* _Jianing Song_, Sep 22 2018 *)
%o A083031 (Magma) [n : n in [0..300] | n mod 12 in [0, 3, 7]]; // _Wesley Ivan Hurt_, Jun 14 2016
%o A083031 (PARI) x='x+O('x^99); concat(0, Vec(x^2*(3+4*x+5*x^2)/((1+x+x^2)*(1-x)^2))) \\ _Jianing Song_, Sep 22 2018
%Y A083031 A guide for some sequences related to modes and chords:
%Y A083031 Modes:
%Y A083031 Lydian mode (F): A083089
%Y A083031 Ionian mode (C): A083026
%Y A083031 Mixolydian mode (G): A083120
%Y A083031 Dorian mode (D): A083033
%Y A083031 Aeolian mode (A): A060107 (raised seventh: A083028)
%Y A083031 Phrygian mode (E): A083034
%Y A083031 Locrian mode (B): A082977
%Y A083031 Chords:
%Y A083031 Major chord: A083030
%Y A083031 Minor chord: this sequence
%Y A083031 Dominant seventh chord: A083032
%K A083031 nonn,easy
%O A083031 1,2
%A A083031 James Ingram (j.ingram(AT)t-online.de), Jun 01 2003