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A083089
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Numbers that are congruent to {0, 2, 4, 6, 7, 9, 11} mod 12.
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16
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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, 43, 45, 47, 48, 50, 52, 54, 55, 57, 59, 60, 62, 64, 66, 67, 69, 71, 72, 74, 76, 78, 79, 81, 83, 84, 86, 88, 90, 91, 93, 95, 96, 98, 100, 102, 103, 105, 107, 108, 110
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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 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.
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LINKS
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FORMULA
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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
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.
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.
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)
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MAPLE
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MATHEMATICA
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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 *)
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PROG
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(Magma) [n : n in [0..150] | n mod 12 in [0, 2, 4, 6, 7, 9, 11]]; // Wesley Ivan Hurt, Jul 20 2016
(PARI) a(n) = 2*(n-1)-2*(n-1)\7; \\ Altug Alkan, Sep 21 2018
(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
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CROSSREFS
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A guide for some sequences related to modes and chords:
Modes:
Lydian mode (F): 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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