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 A319452 Numbers that are congruent to {0, 3, 6, 10} mod 12. 4

%I

%S 0,3,6,10,12,15,18,22,24,27,30,34,36,39,42,46,48,51,54,58,60,63,66,70,

%T 72,75,78,82,84,87,90,94,96,99,102,106,108,111,114,118,120,123,126,

%U 130,132,135,138,142,144,147,150,154,156,159,162,166,168,171,174,178

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

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

%H Jianing Song, <a href="/A319452/b319452.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).

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

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

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

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

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

%t Select[Range[0, 200], MemberQ[{0, 3, 6, 10}, Mod[#, 12]]&]

%t LinearRecurrence[{1, 0, 0, 1, -1}, {0, 3, 6, 10, 12}, 100]

%o (MAGMA) [n : n in [0..150] | n mod 12 in [0, 3, 6, 10]]

%o (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)*(1-x)^2)))

%Y A guide for some sequences related to modes and chords:

%Y Modes:

%Y Lydian mode (F): A083089

%Y Ionian mode (C): A083026

%Y Mixolydian mode (G): A083120

%Y Dorian mode (D): A083033

%Y Aeolian mode (A): A060107 (raised seventh: A083028)

%Y Phrygian mode (E): A083034

%Y Locrian mode (B): A082977

%Y Third chords:

%Y Major chord (F,C,G): A083030

%Y Minor chord (D,A,E): A083031

%Y Diminished chord (B): A319451

%Y Seventh chords:

%Y Major seventh chord (F,C): A319280

%Y Dominant seventh chord (G): A083032

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

%Y Half-diminished seventh chord (B): this sequence

%K nonn,easy

%O 1,2

%A _Jianing Song_, Sep 19 2018

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Last modified February 18 02:57 EST 2020. Contains 332006 sequences. (Running on oeis4.)