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A083033
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Numbers that are congruent to {0, 2, 3, 5, 7, 9, 10} mod 12.
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16
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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, 43, 45, 46, 48, 50, 51, 53, 55, 57, 58, 60, 62, 63, 65, 67, 69, 70, 72, 74, 75, 77, 79, 81, 82, 84, 86, 87, 89, 91, 93, 94, 96, 98, 99, 101, 103, 105, 106, 108, 110, 111
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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FORMULA
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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
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.
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.
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)
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MAPLE
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A083033:= n-> 12*floor((n-1)/7)+[0, 2, 3, 5, 7, 9, 10][((n-1) mod 7)+1]:
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MATHEMATICA
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Select[Range[0, 150], MemberQ[{0, 2, 3, 5, 7, 9, 10}, Mod[#, 12]] &] (* Wesley Ivan Hurt, Jul 20 2016 *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 2, 3, 5, 7, 9, 10, 12}, 70] (* Jianing Song, Sep 22 2018 *)
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PROG
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(Magma) [n : n in [0..150] | n mod 12 in [0, 2, 3, 5, 7, 9, 10]]; // Wesley Ivan Hurt, Jul 20 2016
(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
(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
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CROSSREFS
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A guide for some sequences related to modes and chords:
Modes:
Dorian mode (D): this sequence
Chords:
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KEYWORD
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nonn,easy
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AUTHOR
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James Ingram (j.ingram(AT)t-online.de), Jun 01 2003
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STATUS
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approved
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