A082977 Numbers that are congruent to {0, 1, 3, 5, 6, 8, 10} mod 12.
0, 1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 18, 20, 22, 24, 25, 27, 29, 30, 32, 34, 36, 37, 39, 41, 42, 44, 46, 48, 49, 51, 53, 54, 56, 58, 60, 61, 63, 65, 66, 68, 70, 72, 73, 75, 77, 78, 80, 82, 84, 85, 87, 89, 90, 92, 94, 96, 97, 99, 101, 102, 104, 106, 108, 109, 111
Offset: 1
References
- Robert Fludd, Utriusque Cosmi ... Historia, Oppenheim, 1617-1619.
Links
- Vincenzo Librandi, Table of n, a(n) for n = 1..1000
- Robert Fludd, Page 158 of "Utriusque Cosmi" in Beinecke Rare Book and Manuscript Library Photonegatives Collection.
- Robert Fludd, Larger version of the same image
- Robert Fludd, Utriusque Cosmi, Maioris scilicet et Minoris, metaphysica, physica, atque technica Historia, available as ZIP or PDF download.
- Wikipedia, Robert Fludd
- Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-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): this sequence
Chords:
Major chord: A083030
Minor chord: A083031
Dominant seventh chord: A083032
Programs
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Haskell
a082977 n = a082977_list !! (n-1) a082977_list = [0, 1, 3, 5, 6, 8, 10] ++ map (+ 12) a082977_list -- Reinhard Zumkeller, Jan 07 2014
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Magma
[n : n in [0..150] | n mod 12 in [0, 1, 3, 5, 6, 8, 10]]; // Wesley Ivan Hurt, Jul 19 2016
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Maple
A082977:=n->12*floor(n/7)+[0, 1, 3, 5, 6, 8, 10][(n mod 7)+1]: seq(A082977(n), n=0..100); # Wesley Ivan Hurt, Jul 19 2016
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Mathematica
CoefficientList[Series[x(1 + x + 2*x^4)(1 + x + x^2)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)), {x, 0, 100}], x] (* Vincenzo Librandi, Jan 06 2013 *) fQ[n_] := MemberQ[{0, 1, 3, 5, 6, 8, 10}, Mod[n, 12]]; Select[ Range[0, 111], fQ] (* Robert G. Wilson v, Jan 07 2014 *) LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 1, 3, 5, 6, 8, 10, 12}, 70] (* Jianing Song, Sep 22 2018 *) Floor@Range[0,2^8,12/7] (* Federico Provvedi, Oct 18 2018 *) -
PARI
x='x+O('x^99); concat(0, Vec(x*(1+x+2*x^4)*(1+x+x^2)/((1-x)^2*(1+x+x^2+x^3+x^4+x^5+x^6)))) \\ Jianing Song, Sep 22 2018
Formula
G.f.: x*(1 + x + 2*x^4)*(1 + x + x^2)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)). - R. J. Mathar, Sep 17 2008
From Wesley Ivan Hurt, Jul 19 2016: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.
a(n) = (84*n - 105 - 2*(n mod 7) - 2*((n + 1) mod 7) + 5*((n + 2) mod 7) - 2*((n + 3) mod 7) - 2*((n + 4) mod 7) + 5*((n + 5) mod 7) - 2*((n + 6) mod 7))/49.
a(7k) = 12k - 2, a(7k-1) = 12k - 4, a(7k-2) = 12k - 6, a(7k-3) = 12k - 7, a(7k-4) = 12k - 9, a(7k-5) = 12k - 11, a(7k-6) = 12k - 12. (End)
a(n) = a(n-7) + 12 for n > 7. - Jianing Song, Sep 22 2018
a(n) = floor(12*(n-1)/7). - Federico Provvedi, Oct 18 2018
Comments