A083030 - OEIS

%I #30 Sep 08 2022 08:45:10

%S 0,4,7,12,16,19,24,28,31,36,40,43,48,52,55,60,64,67,72,76,79,84,88,91,

%T 96,100,103,108,112,115,120,124,127,132,136,139,144,148,151,156,160,

%U 163,168,172,175,180,184,187,192,196,199,204,208,211,216,220

%N Numbers that are congruent to {0, 4, 7} mod 12.

%C Key-numbers of the pitches of a major 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*(4 + 3*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) = 4*n - (13 + 2*cos(2*n*Pi/3))/3.

%F a(3k) = 12k - 5, a(3k-1) = 12k - 8, a(3k-2) = 12k - 12. (End)

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

%p A083030:=n->4*n-(13+2*cos(2*n*Pi/3))/3: seq(A083030(n), n=1..100); # _Wesley Ivan Hurt_, Jun 14 2016

%t Select[Range[0,250], MemberQ[{0,4,7}, Mod[#,12]]&] (* _Harvey P. Dale_, Apr 17 2014 *)

%t LinearRecurrence[{1, 0, 1, -1}, {0, 4, 7, 12}, 100] (* _Jianing Song_, Sep 22 2018 *)

%o (Magma) [n : n in [0..300] | n mod 12 in [0, 4, 7]]; // _Wesley Ivan Hurt_, Jun 14 2016

%o (PARI) my(x='x+O('x^99)); concat(0, Vec(x^2*(4+3*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: this sequence

%Y Minor chord: A083031

%Y Dominant seventh chord: A083032

%K nonn,easy

%O 1,2

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