A083032 Numbers that are congruent to {0, 4, 7, 10} mod 12.
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
Offset: 1
Links
- 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).
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
Phrygian mode (E): A083034
Locrian mode (B): A082977
Chords:
Major chord: A083030
Minor chord: A083031
Dominant seventh chord: this sequence
Programs
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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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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
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Maple
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
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 *) -
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
Formula
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)
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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