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A083033
Numbers that are congruent to {0, 2, 3, 5, 7, 9, 10} mod 12.
16
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
OFFSET
1,2
COMMENTS
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.
FORMULA
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
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 - 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)
a(n) = a(n-7) + 12 for n > 7. - Jianing Song, Sep 22 2018
a(n) = floor(3 * (4*n - 3) / 7). - Federico Provvedi, Nov 06 2023
MAPLE
A083033:= n-> 12*floor((n-1)/7)+[0, 2, 3, 5, 7, 9, 10][((n-1) mod 7)+1]:
seq(A083033(n), n=1..100); # Wesley Ivan Hurt, Jul 20 2016
MATHEMATICA
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 *)
Quotient[3 (4#-3), 7] & /@ Range[96] (* Federico Provvedi, Nov 06 2023 *)
PROG
(Magma) [n : n in [0..150] | n mod 12 in [0, 2, 3, 5, 7, 9, 10]]; // Wesley Ivan Hurt, Jul 20 2016
(PARI) a(n)=[-2, 0, 2, 3, 5, 7, 9][n%7+1] + n\7*12 \\ Charles R Greathouse IV, 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
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): this sequence
Aeolian mode (A): A060107 (raised seventh: A083028)
Phrygian mode (E): A083034
Locrian mode (B): A082977
Chords:
Major chord: A083030
Minor chord: A083031
Dominant seventh chord: A083032
Sequence in context: A226249 A076355 A081477 * A022847 A047371 A327492
KEYWORD
nonn,easy
AUTHOR
James Ingram (j.ingram(AT)t-online.de), Jun 01 2003
STATUS
approved