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%I #34 Nov 06 2023 18:13:02
%S 0,2,3,5,7,9,10,12,14,15,17,19,21,22,24,26,27,29,31,33,34,36,38,39,41,
%T 43,45,46,48,50,51,53,55,57,58,60,62,63,65,67,69,70,72,74,75,77,79,81,
%U 82,84,86,87,89,91,93,94,96,98,99,101,103,105,106,108,110,111
%N Numbers that are congruent to {0, 2, 3, 5, 7, 9, 10} mod 12.
%C Key-numbers of the pitches of a Dorian mode scale on a standard chromatic keyboard, with root = 0. A Dorian mode scale can, for example, be played on consecutive white keys of a standard keyboard, starting on the root tone D.
%H Muniru A Asiru, <a href="/A083033/b083033.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^2 + 1)*(2*x^4 + x^3 + 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 - 84 + 5*(n mod 7) - 2*((n + 1) mod 7) - 2*((n + 2) mod 7) - 2*((n + 3) mod 7) + 5*((n + 4) mod 7) - 2*((n + 5) mod 7) - 2*((n + 6) mod 7))/49.
%F a(7k) = 12k - 2, a(7k-1) = 12k - 3, a(7k-2) = 12k - 5, a(7k-3) = 12k - 7, a(7k-4) = 12k - 9, a(7k-5) = 12k - 10, 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(3 * (4*n - 3) / 7). - _Federico Provvedi_, Nov 06 2023
%p A083033:= n-> 12*floor((n-1)/7)+[0, 2, 3, 5, 7, 9, 10][((n-1) mod 7)+1]:
%p seq(A083033(n), n=1..100); # _Wesley Ivan Hurt_, Jul 20 2016
%t Select[Range[0, 150], MemberQ[{0, 2, 3, 5, 7, 9, 10}, Mod[#, 12]] &] (* _Wesley Ivan Hurt_, Jul 20 2016 *)
%t LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 2, 3, 5, 7, 9, 10, 12}, 70] (* _Jianing Song_, Sep 22 2018 *)
%t Quotient[3 (4#-3), 7] & /@ Range[96] (* _Federico Provvedi_, Nov 06 2023 *)
%o (Magma) [n : n in [0..150] | n mod 12 in [0, 2, 3, 5, 7, 9, 10]]; // _Wesley Ivan Hurt_, Jul 20 2016
%o (PARI) a(n)=[-2, 0, 2, 3, 5, 7, 9][n%7+1] + n\7*12 \\ _Charles R Greathouse IV_, Jul 20 2016
%o (PARI) my(x='x+O('x^99)); concat(0, Vec(x^2*(x^2+1)*(2*x^4+x^3+x+2)/((x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2))) \\ _Jianing Song_, Sep 22 2018
%o (GAP) Filtered([0..120],n-> n mod 12=0 or n mod 12=2 or n mod 12=3 or n mod 12=5 or n mod 12=7 or n mod 12=9 or n mod 12=10); # _Muniru A Asiru_, Sep 22 2018
%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): this sequence
%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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