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Numbers that are congruent to {0, 1, 3, 5, 6, 8, 10} mod 12.
18

%I #54 Sep 08 2022 08:45:10

%S 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,

%T 42,44,46,48,49,51,53,54,56,58,60,61,63,65,66,68,70,72,73,75,77,78,80,

%U 82,84,85,87,89,90,92,94,96,97,99,101,102,104,106,108,109,111

%N Numbers that are congruent to {0, 1, 3, 5, 6, 8, 10} mod 12.

%C James Ingram suggests that this (with the initial 0 omitted) is the correct version of Fludd's sequence, rather than A047329. See also A083026.

%C Key-numbers of the pitches of a Hypophrygian mode scale on a standard chromatic keyboard, with root = 0. A Hypophrygian mode scale can, for example, be played on consecutive white keys of a standard keyboard, starting on the root tone B. - James Ingram (j.ingram(AT)t-online.de), Jun 01 2003

%D Robert Fludd, Utriusque Cosmi ... Historia, Oppenheim, 1617-1619.

%H Vincenzo Librandi, <a href="/A082977/b082977.txt">Table of n, a(n) for n = 1..1000</a>

%H Robert Fludd, <a href="http://beinecke.library.yale.edu/dl_crosscollex/photoneg/printoneITEM.asp?pid=39002036242312&amp;iid=3624231">Page 158 of "Utriusque Cosmi"</a> in Beinecke Rare Book and Manuscript Library Photonegatives Collection.

%H Robert Fludd, <a href="http://highway55.library.yale.edu/PHOTONEGIMG/zoom/Z362/z3624231.jpg">Larger version of the same image</a>

%H Robert Fludd, <a href="http://billheidrick.com/Orpd/RFludd/">Utriusque Cosmi, Maioris scilicet et Minoris, metaphysica, physica, atque technica Historia</a>, available as ZIP or PDF download.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Robert_Fludd">Robert Fludd</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,1,-1).

%F 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

%F From _Wesley Ivan Hurt_, Jul 19 2016: (Start)

%F a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.

%F 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.

%F 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)

%F a(n) = a(n-7) + 12 for n > 7. - _Jianing Song_, Sep 22 2018

%F a(n) = floor(12*(n-1)/7). - _Federico Provvedi_, Oct 18 2018

%p 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

%t 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 *)

%t 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 *)

%t LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 1, 3, 5, 6, 8, 10, 12}, 70] (* _Jianing Song_, Sep 22 2018 *)

%t Floor@Range[0,2^8,12/7] (* _Federico Provvedi_, Oct 18 2018 *)

%o (Haskell)

%o a082977 n = a082977_list !! (n-1)

%o a082977_list = [0, 1, 3, 5, 6, 8, 10] ++ map (+ 12) a082977_list

%o -- _Reinhard Zumkeller_, Jan 07 2014

%o (Magma) [n : n in [0..150] | n mod 12 in [0, 1, 3, 5, 6, 8, 10]]; // _Wesley Ivan Hurt_, Jul 19 2016

%o (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

%Y Cf. A047329. Different from A000210.

%Y A guide for some sequences related to modes and chords:

%Y Modes:

%Y Lydian mode (F): A083089

%Y Ionian mode (C): A083026

%Y Mixolydian mode (G): A083120

%Y Dorian mode (D): A083033

%Y Aeolian mode (A): A060107 (raised seventh: A083028)

%Y Phrygian mode (E): A083034

%Y Locrian mode (B): this sequence

%Y Chords:

%Y Major chord: A083030

%Y Minor chord: A083031

%Y Dominant seventh chord: A083032

%K nonn,easy

%O 1,3

%A _N. J. A. Sloane_, May 31 2003