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

%I A319451 #40 Sep 08 2022 08:46:23
%S A319451 0,3,6,12,15,18,24,27,30,36,39,42,48,51,54,60,63,66,72,75,78,84,87,90,
%T A319451 96,99,102,108,111,114,120,123,126,132,135,138,144,147,150,156,159,
%U A319451 162,168,171,174,180,183,186,192,195,198,204,207,210,216,219,222,228
%N A319451 Numbers that are congruent to {0, 3, 6} mod 12; a(n) = 3*floor(4*n/3).
%C A319451 Key-numbers of the pitches of a diminished chord on a standard chromatic keyboard, with root = 0.
%H A319451 Jianing Song, <a href="/A319451/b319451.txt">Table of n, a(n) for n = 0..10000</a>
%H A319451 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,1,-1).
%F A319451 a(n) = a(n-3) + 12 for n > 2.
%F A319451 a(n) = a(n-1) + a(n-3) - a(n-4) for n > 3.
%F A319451 G.f.: 3*(1 + x + 2*x^2)/((1 - x)*(1 - x^3)).
%F A319451 a(n) = 3*A004773(n) = 3*(floor(n/3) + n).
%F A319451 a(n) = 4*n - 1 + sin((Pi/3)*(2*n + 1))/sin(Pi/3). - _Federico Provvedi_, Oct 23 2018
%F A319451 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
%F A319451 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
%p A319451 seq(3*floor(4*n/3),n=0..60); # _Muniru A Asiru_, Oct 24 2018
%t A319451 Select[Range[0, 200], MemberQ[{0, 3, 6}, Mod[#, 12]]&]
%t A319451 LinearRecurrence[{1, 0, 1, -1}, {0, 3, 6, 12}, 100]
%t A319451 Table[4n-1+Sin[Pi/3(2n+1)]/Sin[Pi/3],{n,0,99}] (* _Federico Provvedi_, Oct 23 2018 *)
%o A319451 (Magma) [n : n in [0..150] | n mod 12 in [0, 3, 6]]
%o A319451 (PARI) a(n)=3*(4*n\3)
%o A319451 (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
%o A319451 (Python) for n in range(0,60): print(3*int(4*n/3), end=", ") # _Stefano Spezia_, Dec 07 2018
%Y A319451 Cf. A004773, A047464.
%Y A319451 A guide for some sequences related to modes and chords:
%Y A319451 Modes:
%Y A319451 Lydian mode (F): A083089
%Y A319451 Ionian mode (C): A083026
%Y A319451 Mixolydian mode (G): A083120
%Y A319451 Dorian mode (D): A083033
%Y A319451 Aeolian mode (A): A060107 (raised seventh: A083028)
%Y A319451 Phrygian mode (E): A083034
%Y A319451 Locrian mode (B): A082977
%Y A319451 Third chords:
%Y A319451 Major chord (F,C,G): A083030
%Y A319451 Minor chord (D,A,E): A083031
%Y A319451 Diminished chord (B): this sequence
%Y A319451 Seventh chords:
%Y A319451 Major seventh chord (F,C): A319280
%Y A319451 Dominant seventh chord (G): A083032
%Y A319451 Minor seventh chord (D,A,E): A319279
%Y A319451 Half-diminished seventh chord (B): A319452
%K A319451 nonn,easy
%O A319451 0,2
%A A319451 _Jianing Song_, Sep 19 2018