A319451 Numbers that are congruent to {0, 3, 6} mod 12; a(n) = 3*floor(4*n/3).

0, 3, 6, 12, 15, 18, 24, 27, 30, 36, 39, 42, 48, 51, 54, 60, 63, 66, 72, 75, 78, 84, 87, 90, 96, 99, 102, 108, 111, 114, 120, 123, 126, 132, 135, 138, 144, 147, 150, 156, 159, 162, 168, 171, 174, 180, 183, 186, 192, 195, 198, 204, 207, 210, 216, 219, 222, 228

Offset: 0

Author

Jianing Song, Sep 19 2018

Keywords

• nonn
• easy

Comments

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

Links

• Jianing Song, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Crossrefs

Cf. A004773, A047464.

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): this sequence

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): A319452

Programs

GAP

Filtered([0..230],n->n mod 12 = 0 or n mod 12 = 3 or n mod 12 = 6); # Muniru A Asiru, Oct 24 2018

Magma

[n : n in [0..150] | n mod 12 in [0, 3, 6]]

Maple

seq(3*floor(4*n/3),n=0..60); # Muniru A Asiru, Oct 24 2018

Mathematica

Select[Range[0, 200], MemberQ[{0, 3, 6}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 1, -1}, {0, 3, 6, 12}, 100]
Table[4n-1+Sin[Pi/3(2n+1)]/Sin[Pi/3],{n,0,99}] (* Federico Provvedi, Oct 23 2018 *)

PARI

a(n)=3*(4*n\3)

Python

for n in range(0,60): print(3*int(4*n/3), end=", ") # Stefano Spezia, Dec 07 2018

Formula

a(n) = a(n-3) + 12 for n > 2.

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

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

a(n) = 3*A004773(n) = 3*(floor(n/3) + n).

a(n) = 4*n - 1 + sin((Pi/3)*(2*n + 1))/sin(Pi/3). - Federico Provvedi, Oct 23 2018

E.g.f.: (3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/(3*exp(x/2)) - exp(x)*(1 - 4*x). - Franck Maminirina Ramaharo, Nov 27 2018

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