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A083031 Numbers that are congruent to {0, 3, 7} mod 12. 15

%I

%S 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 96,99,103,108,111,115,120,123,127,132,135,139,144,147,151,156,159,

%U 163,168,171,175,180,183,187,192,195,199,204,207,211,216,219

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

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

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).

%F G.f.: x^2*(3 + 4*x + 5*x^2)/((1 + x + x^2)*(1 - x)^2). - _R. J. Mathar_, Oct 08 2011

%F From _Wesley Ivan Hurt_, Jun 14 2016: (Start)

%F a(n) = a(n-1) + a(n-3) - a(n-4) for n > 4.

%F a(n) = (12*n - 14 - cos(2*n*Pi/3) + sqrt(3)*sin(2*n*Pi/3))/3.

%F a(3k) = 12k - 5, a(3k-1) = 12k - 9, a(3k-2) = 12k - 12. (End)

%F a(n) = a(n-3) + 12 for n > 3. - _Jianing Song_, Sep 22 2018

%p 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 Select[Range[0, 400], MemberQ[{0, 3, 7}, Mod[#, 12]] &] (* _Wesley Ivan Hurt_, Jun 14 2016 *)

%t LinearRecurrence[{1, 0, 1, -1}, {0, 3, 7, 12}, 100] (* _Jianing Song_, Sep 22 2018 *)

%o (MAGMA) [n : n in [0..300] | n mod 12 in [0, 3, 7]]; // _Wesley Ivan Hurt_, Jun 14 2016

%o (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 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 Chords:

%Y Major chord: A083030

%Y Minor chord: this sequence

%Y Dominant seventh chord: A083032

%K nonn,easy

%O 1,2

%A James Ingram (j.ingram(AT)t-online.de), Jun 01 2003

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Last modified August 10 08:35 EDT 2020. Contains 336368 sequences. (Running on oeis4.)