A083089 Numbers that are congruent to {0, 2, 4, 6, 7, 9, 11} mod 12.

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

Offset

1,2

Comments

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.

Links

Vincenzo Librandi, 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^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

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 - 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)

a(n) = 2*n - 2 - floor(2*(n - 1)/7). - Wesley Ivan Hurt, Sep 29 2017

a(n) = a(n-7) + 12 for n > 7. - Jianing Song, Sep 22 2018

Maple

A083089:=n->12*floor(n/7)+[0, 2, 4, 6, 7, 9, 11][(n mod 7)+1]: seq(A083089(n), n=0..100); # Wesley Ivan Hurt, Jul 20 2016

Mathematica

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 *)

Prog

(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

Crossrefs

A guide for some sequences related to modes and chords:

Modes:

Lydian mode (F): this sequence

Ionian mode (C): A083026

Mixolydian mode (G): A083120

Dorian mode (D): A083033

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: A083088 A080755 A372779 * A136617 A275814 A285376

Adjacent sequences: A083086 A083087 A083088 * A083090 A083091 A083092

Keyword

nonn,easy

Author

James Ingram (j.ingram(AT)t-online.de), Jun 01 2003

Status

approved