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A083034
Numbers that are congruent to {0, 1, 3, 5, 7, 8, 10} mod 12.
15
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, 43, 44, 46, 48, 49, 51, 53, 55, 56, 58, 60, 61, 63, 65, 67, 68, 70, 72, 73, 75, 77, 79, 80, 82, 84, 85, 87, 89, 91, 92, 94, 96, 97, 99, 101, 103, 104, 106, 108, 109, 111
OFFSET
1,3
COMMENTS
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.
FORMULA
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
From Wesley Ivan Hurt, Jul 20 2016: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.
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.
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)
a(n) = a(n-7) + 12 for n > 7. - Jianing Song, Sep 22 2018
a(n) = floor((12*n - 11) / 7). - Federico Provvedi, Nov 06 2023
MAPLE
A083034:= n-> 12*floor((n-1)/7)+[0, 1, 3, 5, 7, 8, 10][((n-1) mod 7)+1]:
seq(A083034(n), n=1..100); # Wesley Ivan Hurt, Jul 20 2016
MATHEMATICA
Select[Range[0, 150], MemberQ[{0, 1, 3, 5, 7, 8, 10}, Mod[#, 12]] &] (* Wesley Ivan Hurt, Jul 20 2016 *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 1, 3, 5, 7, 8, 10, 12}, 70] (* Jianing Song, Sep 22 2018 *)
Quotient[12 # - 11, 7] & /@ Range[96] (* Federico Provvedi, Nov 06 2023 *)
PROG
(Magma) [n : n in [0..150] | n mod 12 in [0, 1, 3, 5, 7, 8, 10]]; // Wesley Ivan Hurt, Jul 20 2016
(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
CROSSREFS
A guide for some sequences related to modes and chords:
Modes:
Lydian mode (F): A083089
Ionian mode (C): A083026
Mixolydian mode (G): A083120
Dorian mode (D): A083033
Aeolian mode (A): A060107 (raised seventh: A083028)
Phrygian mode (E): this sequence
Locrian mode (B): A082977
Chords:
Major chord: A083030
Minor chord: A083031
Dominant seventh chord: A083032
Sequence in context: A285074 A186219 A185050 * A213908 A247514 A144077
KEYWORD
nonn,easy
AUTHOR
James Ingram (j.ingram(AT)t-online.de), Jun 01 2003
STATUS
approved