A060107 - OEIS

%I #36 Sep 08 2022 08:45:03

%S 0,2,3,5,7,8,10,12,14,15,17,19,20,22,24,26,27,29,31,32,34,36,38,39,41,

%T 43,44,46,48,50,51,53,55,56,58,60,62,63,65,67,68,70,72,74,75,77,79,80,

%U 82,84,86,87,89,91,92,94,96,98,99,101,103,104,106,108,110,111,113,115

%N Numbers that are congruent to {0, 2, 3, 5, 7, 8, 10} mod 12. The ivory keys on a piano, start with A0 = the 0th key.

%C More precisely, the key-numbers of the pitches of a minor scale on a standard chromatic keyboard, with root = 0 and flat seventh.

%C Also key-numbers of the pitches of an Aeolian mode scale on a standard chromatic keyboard, with root = 0. An Aeolian mode scale can, for example, be played on consecutive white keys of a standard keyboard, starting on the root tone A.

%C A piano sequence since if a(n) < 88 then A059620(a(n)) = 0.

%H Vincenzo Librandi, <a href="/A060107/b060107.txt">Table of n, a(n) for n = 1..2000</a>

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,0,0,0,1,-1).

%F a(n) = a(n-7) + 12 for n > 7.

%F a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.

%F G.f.: x^2*(2 + x + 2*x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^6)/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x - 1)^2). - _R. J. Mathar_, Oct 08 2011

%F From _Wesley Ivan Hurt_, Jul 20 2016: (Start)

%F a(n) = (84*n - 91 - 2*(n mod 7) + 5*((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.

%F 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 - 10, a(7k-6) = 12k - 12. (End)

%F a(n) = A081031(n) - 1 for 1 <= n <= 36. - _Jianing Song_, Oct 14 2019

%p A060107:=n->12*floor(n/7)+[0, 2, 3, 5, 7, 8, 10][(n mod 7)+1]: seq(A060107(n), n=0..100); # _Wesley Ivan Hurt_, Jul 20 2016

%t Select[Range[0,120], MemberQ[{0,2,3,5,7,8,10}, Mod[#,12]]&] (* or *) LinearRecurrence[{1,0,0,0,0,0,1,-1}, {0,2,3,5,7,8,10,12}, 70] (* _Harvey P. Dale_, Nov 10 2011 *)

%o (Magma) [n : n in [0..150] | n mod 12 in [0, 2, 3, 5, 7, 8, 10]]; // _Wesley Ivan Hurt_, Jul 20 2016

%o (PARI) x='x+O('x^99); concat(0, Vec(x^2*(2+x+2*x^2+2*x^3+x^4+2*x^5+2*x^6)/((x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2))) \\ _Jianing Song_, Sep 22 2018

%Y Cf. A059620, A081031. Complement of A060106.

%Y A guide for some sequences related to modes and chords:

%Y Modes:

%Y Lydian mode (F): A083089

%Y Ionian mode (C): A083026

%Y Mixolydian mode (G): A083120

%Y Dorian mode (D): A083033

%Y Aeolian mode (A): this sequence (raised seventh: A083028)

%Y Phrygian mode (E): A083034

%Y Locrian mode (B): A082977

%Y Chords:

%Y Major chord: A083030

%Y Minor chord: A083031

%Y Dominant seventh chord: A083032

%K easy,nonn

%O 1,2

%A _Henry Bottomley_, Feb 27 2001