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%I #34 Sep 08 2022 08:45:10
%S 0,2,4,6,7,9,11,12,14,16,18,19,21,23,24,26,28,30,31,33,35,36,38,40,42,
%T 43,45,47,48,50,52,54,55,57,59,60,62,64,66,67,69,71,72,74,76,78,79,81,
%U 83,84,86,88,90,91,93,95,96,98,100,102,103,105,107,108,110
%N Numbers that are congruent to {0, 2, 4, 6, 7, 9, 11} mod 12.
%C Key-numbers of the pitches of a Lydian mode scale on a standard chromatic keyboard, with root = 0. A Lydian mode scale can, for example, be played on consecutive white keys of a standard keyboard, starting on the root tone F.
%H Vincenzo Librandi, <a href="/A083089/b083089.txt">Table of n, a(n) for n = 1..2000</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^2*(x^4 + x^3 + 2)*(1 + x + x^2)/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x - 1)^2). - _R. J. Mathar_, Oct 08 2011
%F From _Wesley Ivan Hurt_, Jul 20 2016: (Start)
%F a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.
%F a(n) = (84*n - 63 - 2*(n mod 7) - 2*((n + 1) mod 7) + 5*((n + 2) mod 7) - 2*((n + 3) mod 7) - 2*((n + 4) mod 7) - 2*((n + 5) mod 7) + 5*((n + 6) mod 7))/49.
%F a(7k) = 12k - 1, a(7k - 1) = 12k - 3, a(7k-2) = 12k - 5, a(7k-3) = 12k - 6, a(7k-4) = 12k - 8, a(7k-5) = 12k - 10, a(7k-6) = 12k - 12. (End)
%F a(n) = 2*n - 2 - floor(2*(n - 1)/7). - _Wesley Ivan Hurt_, Sep 29 2017
%F a(n) = a(n-7) + 12 for n > 7. - _Jianing Song_, Sep 22 2018
%p A083089:=n->12*floor(n/7)+[0, 2, 4, 6, 7, 9, 11][(n mod 7)+1]: seq(A083089(n), n=0..100); # _Wesley Ivan Hurt_, Jul 20 2016
%t Select[Range[0,200],MemberQ[{0,2,4,6,7,9,11},Mod[#,12]]&] (* or *) LinearRecurrence[{1,0,0,0,0,0,1,-1},{0,2,4,6,7,9,11,12},90] (* _Harvey P. Dale_, Mar 29 2016 *)
%o (Magma) [n : n in [0..150] | n mod 12 in [0, 2, 4, 6, 7, 9, 11]]; // _Wesley Ivan Hurt_, Jul 20 2016
%o (PARI) a(n) = 2*(n-1)-2*(n-1)\7; \\ _Altug Alkan_, Sep 21 2018
%o (PARI) x='x+O('x^99); concat(0, Vec(x^2*(x^4+x^3+2)*(1+x+x^2)/((x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2))) \\ _Jianing Song_, Sep 22 2018
%Y A guide for some sequences related to modes and chords:
%Y Modes:
%Y Lydian mode (F): this sequence
%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): A082977
%Y Chords:
%Y Major chord: A083030
%Y Minor chord: A083031
%Y Dominant seventh chord: A083032
%K nonn,easy
%O 1,2
%A James Ingram (j.ingram(AT)t-online.de), Jun 01 2003
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