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.
LINKS
Muniru A Asiru, Table of n, a(n) for n = 1..2000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).
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
Phrygian mode (E): A083034
Locrian mode (B): A082977
Chords:
Major chord: A083030
Minor chord: A083031
Dominant seventh chord: A083032
KEYWORD
nonn,easy
AUTHOR
James Ingram (j.ingram(AT)t-online.de), Jun 01 2003
STATUS
approved