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

%I A319452 #20 Sep 08 2022 08:46:23
%S A319452 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 A319452 72,75,78,82,84,87,90,94,96,99,102,106,108,111,114,118,120,123,126,
%U A319452 130,132,135,138,142,144,147,150,154,156,159,162,166,168,171,174,178
%N A319452 Numbers that are congruent to {0, 3, 6, 10} mod 12.
%C A319452 Key-numbers of the pitches of a half-diminished chord on a standard chromatic keyboard, with root = 0.
%H A319452 Jianing Song, <a href="/A319452/b319452.txt">Table of n, a(n) for n = 1..10000</a>
%H A319452 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F A319452 a(n) = a(n-4) + 12 for n > 4.
%F A319452 a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
%F A319452 G.f.: x^2*(3 + 3*x + 4*x^2 + 2*x^3)/((1 + x)*(1 + x^2)*(1 - x)^2).
%F A319452 a(n) = (12*n - 11 + (-1)^n + 2*cos(Pi*n/2))/4.
%F A319452 E.g.f.: ((6*x - 5)*cosh(x) + (6*x - 6)*sinh(x) + cos(x) + 4)/2.
%F A319452 Sum_{n>=2} (-1)^n/a(n) = log(12)/8 - (sqrt(3)-1)*Pi/24. - _Amiram Eldar_, Dec 30 2021
%t A319452 Select[Range[0, 200], MemberQ[{0, 3, 6, 10}, Mod[#, 12]]&]
%t A319452 LinearRecurrence[{1, 0, 0, 1, -1}, {0, 3, 6, 10, 12}, 100]
%o A319452 (Magma) [n : n in [0..150] | n mod 12 in [0, 3, 6, 10]]
%o A319452 (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)))
%Y A319452 A guide for some sequences related to modes and chords:
%Y A319452 Modes:
%Y A319452 Lydian mode (F): A083089
%Y A319452 Ionian mode (C): A083026
%Y A319452 Mixolydian mode (G): A083120
%Y A319452 Dorian mode (D): A083033
%Y A319452 Aeolian mode (A): A060107 (raised seventh: A083028)
%Y A319452 Phrygian mode (E): A083034
%Y A319452 Locrian mode (B): A082977
%Y A319452 Third chords:
%Y A319452 Major chord (F,C,G): A083030
%Y A319452 Minor chord (D,A,E): A083031
%Y A319452 Diminished chord (B): A319451
%Y A319452 Seventh chords:
%Y A319452 Major seventh chord (F,C): A319280
%Y A319452 Dominant seventh chord (G): A083032
%Y A319452 Minor seventh chord (D,A,E): A319279
%Y A319452 Half-diminished seventh chord (B): this sequence
%K A319452 nonn,easy
%O A319452 1,2
%A A319452 _Jianing Song_, Sep 19 2018