A319280 Numbers that are congruent to {0, 4, 7, 11} mod 12.

0, 4, 7, 11, 12, 16, 19, 23, 24, 28, 31, 35, 36, 40, 43, 47, 48, 52, 55, 59, 60, 64, 67, 71, 72, 76, 79, 83, 84, 88, 91, 95, 96, 100, 103, 107, 108, 112, 115, 119, 120, 124, 127, 131, 132, 136, 139, 143, 144, 148, 151, 155, 156, 160, 163, 167, 168, 172, 175, 179

Offset

1,2

Comments

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

Links

Jianing Song, Table of n, a(n) for n = 1..10000

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

Formula

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

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

G.f.: x^2*(4 + 3*x + 4*x^2 + x^3)/((1 + x)*(1 + x^2)*(1 - x)^2).

a(n) = (6*n - 4 + (-1)^n + sqrt(2)*cos(Pi*n/2 + Pi/4))/2.

E.g.f.: ((6*x - 3)*cosh(x) + (6*x - 5)*sinh(x) + sqrt(2)*cos(x + Pi/4) + 2)/2.

Sum_{n>=2} (-1)^n/a(n) = log(3)/8 + log(2+sqrt(3))/(2*sqrt(3)) - 5*sqrt(3)*Pi/72. - _Amiram Eldar_, Dec 30 2021

Mathematica

Select[Range[0, 200], MemberQ[{0, 4, 7, 11}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 4, 7, 11, 12}, 100]

Prog

(Magma) [n : n in [0..150] | n mod 12 in [0, 4, 7, 11]]
(PARI) my(x='x+O('x^99)); concat(0, Vec(x^2*(4+3*x+4*x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2)))

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

Third chords:

Major chord (F,C,G): A083030

Minor chord (D,A,E): A083031

Diminished chord (B): A319451

Seventh chords:

Major seventh chord (F,C): this sequence

Dominant seventh chord (G): A083032

Minor seventh chord (D,A,E): A319279

Half-diminished seventh chord (B): A319452

Sequence in context: A164888 A023985 A023979 * A214975 A345347 A310720

Adjacent sequences: A319277 A319278 A319279 * A319281 A319282 A319283

Keyword

nonn,easy

Author

_Jianing Song_, Sep 16 2018

Status

approved