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%I #35 Nov 06 2023 18:12:44
%S 0,1,3,5,7,8,10,12,13,15,17,19,20,22,24,25,27,29,31,32,34,36,37,39,41,
%T 43,44,46,48,49,51,53,55,56,58,60,61,63,65,67,68,70,72,73,75,77,79,80,
%U 82,84,85,87,89,91,92,94,96,97,99,101,103,104,106,108,109,111
%N Numbers that are congruent to {0, 1, 3, 5, 7, 8, 10} mod 12.
%C Key-numbers of the pitches of a Phrygian mode scale on a standard chromatic keyboard, with root = 0. A Phrygian mode scale can, for example, be played on consecutive white keys of a standard keyboard, starting on the root tone E.
%H Vincenzo Librandi, <a href="/A083034/b083034.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 + 1)*(2*x^5 + x^3 + x^2 + x + 1)/((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 - 98 - 2*(n mod 7) + 5*((n + 1) mod 7) - 2*((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 - 5, 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 - 11) / 7). - _Federico Provvedi_, Nov 06 2023
%p A083034:= n-> 12*floor((n-1)/7)+[0, 1, 3, 5, 7, 8, 10][((n-1) mod 7)+1]:
%p seq(A083034(n), n=1..100); # _Wesley Ivan Hurt_, Jul 20 2016
%t Select[Range[0, 150], MemberQ[{0, 1, 3, 5, 7, 8, 10}, Mod[#, 12]] &] (* _Wesley Ivan Hurt_, Jul 20 2016 *)
%t LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 1, 3, 5, 7, 8, 10, 12}, 70] (* _Jianing Song_, Sep 22 2018 *)
%t Quotient[12 # - 11, 7] & /@ Range[96] (* _Federico Provvedi_, Nov 06 2023 *)
%o (Magma) [n : n in [0..150] | n mod 12 in [0, 1, 3, 5, 7, 8, 10]]; // _Wesley Ivan Hurt_, Jul 20 2016
%o (PARI) my(x='x+O('x^99)); concat(0, Vec(x^2*(x+1)*(2*x^5+x^3+x^2+x+1)/((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): 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): this sequence
%Y Locrian mode (B): A082977
%Y Chords:
%Y Major chord: A083030
%Y Minor chord: A083031
%Y Dominant seventh chord: A083032
%K nonn,easy
%O 1,3
%A James Ingram (j.ingram(AT)t-online.de), Jun 01 2003
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