

A319452


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


4



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
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OFFSET

1,2


COMMENTS

Keynumbers of the pitches of a halfdiminished 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(n4) + 12 for n > 4.
a(n) = a(n1) + a(n4)  a(n5) 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.


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) x='x+O('x^99); concat(0, Vec(x^2*(3+3*x+4*x^2+2*x^3)/((1+x)*(1+x^2)*(1x)^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
Halfdiminished seventh chord (B): this sequence
Sequence in context: A310043 A316325 A114981 * A187744 A007960 A280243
Adjacent sequences: A319449 A319450 A319451 * A319453 A319454 A319455


KEYWORD

nonn,easy


AUTHOR

Jianing Song, Sep 19 2018


STATUS

approved



