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

0, 3, 7, 12, 15, 19, 24, 27, 31, 36, 39, 43, 48, 51, 55, 60, 63, 67, 72, 75, 79, 84, 87, 91, 96, 99, 103, 108, 111, 115, 120, 123, 127, 132, 135, 139, 144, 147, 151, 156, 159, 163, 168, 171, 175, 180, 183, 187, 192, 195, 199, 204, 207, 211, 216, 219

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..56.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Formula

G.f.: x^2*(3 + 4*x + 5*x^2)/((1 + x + x^2)*(1 - x)^2). - R. J. Mathar, Oct 08 2011

From Wesley Ivan Hurt, Jun 14 2016: (Start)

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

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

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

a(n) = a(n-3) + 12 for n > 3. - Jianing Song, Sep 22 2018

Maple

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

Mathematica

Select[Range[0, 400], MemberQ[{0, 3, 7}, Mod[#, 12]] &] (* Wesley Ivan Hurt, Jun 14 2016 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 3, 7, 12}, 100] (* Jianing Song, Sep 22 2018 *)

Prog

(Magma) [n : n in [0..300] | n mod 12 in [0, 3, 7]]; // Wesley Ivan Hurt, Jun 14 2016
(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

Crossrefs

A guide for some sequences related to modes and chords:

Modes:

Lydian mode (F): A083089

Ionian mode (C): A083026

Mixolydian mode (G): A083120

Dorian mode (D): A083033

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

Phrygian mode (E): A083034

Locrian mode (B): A082977

Chords:

Major chord: A083030

Minor chord: this sequence

Dominant seventh chord: A083032

Sequence in context: A310227 A310228 A310229 * A189927 A189403 A022805

Adjacent sequences: A083028 A083029 A083030 * A083032 A083033 A083034

Keyword

nonn,easy

Author

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

Status

approved