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A083032
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Numbers that are congruent to {0, 4, 7, 10} mod 12.
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16
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0, 4, 7, 10, 12, 16, 19, 22, 24, 28, 31, 34, 36, 40, 43, 46, 48, 52, 55, 58, 60, 64, 67, 70, 72, 76, 79, 82, 84, 88, 91, 94, 96, 100, 103, 106, 108, 112, 115, 118, 120, 124, 127, 130, 132, 136, 139, 142, 144, 148, 151, 154, 156, 160, 163, 166, 168, 172
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OFFSET
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1,2
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COMMENTS
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Key-numbers of the pitches of a dominant seventh chord on a standard chromatic keyboard, with root = 0.
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LINKS
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G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).
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FORMULA
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G.f.: x^2*(4 + 3*x + 3*x^2 + 2*x^3)/((1 + x)*(1 + x^2)*(1 - x)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 19 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
a(n) = (12*n - 9 + (-1)^n + (-1)^((n+1)/2) + (-1)^(-(n+1)/2))/4. (End)
a(2k) = A016957(k-1) for k > 0, a(2k-1) = A272975(k). - Wesley Ivan Hurt, Jun 01 2016
E.g.f.: (4 - sin(x) + (6*x - 5)*sinh(x) + (6*x - 4)*cosh(x))/2. - Ilya Gutkovskiy, Jun 01 2016
From Jianing Song, Sep 22 2018: (Start)
a(n) = (12*n - 9 + (-1)^n - 2*sin(n*Pi/2))/4.
a(n) = a(n-4) + 12 for n > 4. (End)
Sum_{n>=2} (-1)^n/a(n) = log(3)/8 - log(2)/12 + sqrt(3)*log(sqrt(3)+2)/12 - (5*sqrt(3)-6)*Pi/72. - Amiram Eldar, Dec 31 2021
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MAPLE
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A083032:=n->(12*n-9+(-1)^n+(-1)^((n+1)/2)+(-1)^(-(n+1)/2))/4: seq(A083032(n), n=1..100); # Wesley Ivan Hurt, May 19 2016
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MATHEMATICA
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Select[Range[0, 200], MemberQ[{0, 4, 7, 10}, Mod[#, 12]]&] (* Harvey P. Dale, Sep 13 2011 *)
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 4, 7, 10, 12}, 100] (* G. C. Greubel, Jun 01 2016 *)
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PROG
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(Magma) [(12*n-9+(-1)^n+(-1)^((n+1) div 2)+(-1)^(-(n+1) div 2))/4: n in [1..100]]; // Wesley Ivan Hurt, May 19 2016
(PARI) my(x='x+O('x^99)); concat(0, Vec(x^2*(4+3*x+3*x^2+2*x^3)/((1+x)*(1+x^2)*(1-x)^2))) \\ Altug Alkan, Sep 21 2018
(GAP) Filtered([0..200], n-> n mod 12=0 or n mod 12=4 or n mod 12=7 or n mod 12=10); # Muniru A Asiru, Sep 22 2018
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CROSSREFS
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Bisections: A016957, A272975.
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
Chords:
Major chord: A083030
Minor chord: A083031
Dominant seventh chord: this sequence
Sequence in context: A310676 A320929 A104280 * A284933 A020965 A065713
Adjacent sequences: A083029 A083030 A083031 * A083033 A083034 A083035
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KEYWORD
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nonn,easy
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AUTHOR
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James Ingram (j.ingram(AT)t-online.de), Jun 01 2003
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STATUS
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approved
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