A083032 Numbers that are congruent to {0, 4, 7, 10} mod 12.

0, 4, 7, 10, 12, 16, 19, 22, 24, 28, 31, 34, 36, 40, 43, 46, 48, 52, 55, 58, 60, 64, 67, 70, 72, 76, 79, 82, 84, 88, 91, 94, 96, 100, 103, 106, 108, 112, 115, 118, 120, 124, 127, 130, 132, 136, 139, 142, 144, 148, 151, 154, 156, 160, 163, 166, 168, 172

Offset

1,2

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

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

Formula

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

From Wesley Ivan Hurt, May 19 2016: (Start)

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

a(n) = (12*n - 9 + (-1)^n + (-1)^((n+1)/2) + (-1)^(-(n+1)/2))/4. (End)

a(2k) = A016957(k-1) for k > 0, a(2k-1) = A272975(k). - Wesley Ivan Hurt, Jun 01 2016

E.g.f.: (4 - sin(x) + (6*x - 5)*sinh(x) + (6*x - 4)*cosh(x))/2. - Ilya Gutkovskiy, Jun 01 2016

From Jianing Song, Sep 22 2018: (Start)

a(n) = (12*n - 9 + (-1)^n - 2*sin(n*Pi/2))/4.

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

Sum_{n>=2} (-1)^n/a(n) = log(3)/8 - log(2)/12 + sqrt(3)*log(sqrt(3)+2)/12 - (5*sqrt(3)-6)*Pi/72. - Amiram Eldar, Dec 31 2021

Maple

A083032:=n->(12*n-9+(-1)^n+(-1)^((n+1)/2)+(-1)^(-(n+1)/2))/4: seq(A083032(n), n=1..100); # Wesley Ivan Hurt, May 19 2016

Mathematica

Select[Range[0, 200], MemberQ[{0, 4, 7, 10}, Mod[#, 12]]&] (* Harvey P. Dale, Sep 13 2011 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 4, 7, 10, 12}, 100] (* G. C. Greubel, Jun 01 2016 *)

Prog

(Magma) [(12*n-9+(-1)^n+(-1)^((n+1) div 2)+(-1)^(-(n+1) div 2))/4: n in [1..100]]; // Wesley Ivan Hurt, May 19 2016
(PARI) my(x='x+O('x^99)); concat(0, Vec(x^2*(4+3*x+3*x^2+2*x^3)/((1+x)*(1+x^2)*(1-x)^2))) \\ Altug Alkan, Sep 21 2018
(GAP) Filtered([0..200], n-> n mod 12=0 or n mod 12=4 or n mod 12=7 or n mod 12=10); # Muniru A Asiru, Sep 22 2018

Crossrefs

Bisections: A016957, A272975.

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: A083031

Dominant seventh chord: this sequence

Sequence in context: A310676 A320929 A104280 * A284933 A020965 A065713

Adjacent sequences: A083029 A083030 A083031 * A083033 A083034 A083035

Keyword

nonn,easy

Author

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

Status

approved