A319452 Numbers that are congruent to {0, 3, 6, 10} mod 12.

0, 3, 6, 10, 12, 15, 18, 22, 24, 27, 30, 34, 36, 39, 42, 46, 48, 51, 54, 58, 60, 63, 66, 70, 72, 75, 78, 82, 84, 87, 90, 94, 96, 99, 102, 106, 108, 111, 114, 118, 120, 123, 126, 130, 132, 135, 138, 142, 144, 147, 150, 154, 156, 159, 162, 166, 168, 171, 174, 178

Offset

1,2

Comments

Key-numbers of the pitches of a half-diminished 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*(3 + 3*x + 4*x^2 + 2*x^3)/((1 + x)*(1 + x^2)*(1 - x)^2).

a(n) = (12*n - 11 + (-1)^n + 2*cos(Pi*n/2))/4.

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

Sum_{n>=2} (-1)^n/a(n) = log(12)/8 - (sqrt(3)-1)*Pi/24. - Amiram Eldar, Dec 30 2021

Mathematica

Select[Range[0, 200], MemberQ[{0, 3, 6, 10}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 3, 6, 10, 12}, 100]

Prog

(Magma) [n : n in [0..150] | n mod 12 in [0, 3, 6, 10]]
(PARI) my(x='x+O('x^99)); concat(0, Vec(x^2*(3+3*x+4*x^2+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): A319280

Dominant seventh chord (G): A083032

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

Half-diminished seventh chord (B): this sequence

Sequence in context: A310043 A316325 A114981 * A187744 A007960 A339213

Adjacent sequences: A319449 A319450 A319451 * A319453 A319454 A319455

Keyword

nonn,easy

Author

Jianing Song, Sep 19 2018

Status

approved