A082977 -id:A082977 - cp's OEIS frontend

A083042 Duplicate of A082977.

0, 1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 18, 20, 22, 24, 25, 27, 29, 30, 32, 34, 36, 37, 39

Offset: 0

dead

A060107 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.

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, 43, 44, 46, 48, 50, 51, 53, 55, 56, 58, 60, 62, 63, 65, 67, 68, 70, 72, 74, 75, 77, 79, 80, 82, 84, 86, 87, 89, 91, 92, 94, 96, 98, 99, 101, 103, 104, 106, 108, 110, 111, 113, 115

Offset: 1

Henry Bottomley, Feb 27 2001

More precisely, the key-numbers of the pitches of a minor scale on a standard chromatic keyboard, with root = 0 and flat seventh.
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.
A piano sequence since if a(n) < 88 then A059620(a(n)) = 0.

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).

Cf. A059620, A081031. Complement of A060106.
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): this sequence (raised seventh: A083028)
Phrygian mode (E): A083034
Locrian mode (B): A082977
Chords:
Major chord: A083030
Minor chord: A083031
Dominant seventh chord: A083032

[n : n in [0..150] | n mod 12 in [0, 2, 3, 5, 7, 8, 10]]; // Wesley Ivan Hurt, Jul 20 2016

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

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 *)

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

a(n) = a(n-7) + 12 for n > 7.
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.
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
From Wesley Ivan Hurt, Jul 20 2016: (Start)
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.
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)
a(n) = A081031(n) - 1 for 1 <= n <= 36. - Jianing Song, Oct 14 2019

A083026 Numbers that are congruent to {0, 2, 4, 5, 7, 9, 11} mod 12.

Original entry on oeis.org

0, 2, 4, 5, 7, 9, 11, 12, 14, 16, 17, 19, 21, 23, 24, 26, 28, 29, 31, 33, 35, 36, 38, 40, 41, 43, 45, 47, 48, 50, 52, 53, 55, 57, 59, 60, 62, 64, 65, 67, 69, 71, 72, 74, 76, 77, 79, 81, 83, 84, 86, 88, 89, 91, 93, 95, 96, 98, 100, 101, 103, 105, 107, 108, 110

Offset: 1

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

Key-numbers of the pitches of a major scale on a standard chromatic keyboard, with root = 0.
Also key-numbers of the pitches of an Ionian mode scale on a standard chromatic keyboard, with root = 0. An Ionian mode scale can, for example, be played on consecutive white keys of a standard keyboard, starting on the root tone C.
Cumulative sum of A291454. - Halfdan Skjerning, Aug 30 2017

G. C. Greubel, Table of n, a(n) for n = 1..1000
Wikipedia, Scale Music
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A291454.
A guide for some sequences related to modes and chords:
Modes:
Lydian mode (F): A083089
Ionian mode (C): this sequence
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

[n : n in [0..150] | n mod 12 in [0, 2, 4, 5, 7, 9, 11]]; // Wesley Ivan Hurt, Jul 20 2016

A083026:=n->12*floor(n/7)+[0, 2, 4, 5, 7, 9, 11][(n mod 7)+1]: seq(A083026(n), n=0..100); # Wesley Ivan Hurt, Jul 20 2016

Select[Range[0, 150], MemberQ[{0, 2, 4, 5, 7, 9, 11}, Mod[#, 12]] &] (* Wesley Ivan Hurt, Jul 20 2016 *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 2, 4, 5, 7, 9, 11, 12}, 70] (* Jianing Song, Sep 22 2018 *)
Quotient[12*Range[60], 7] - 1 (* Federico Provvedi, Sep 10 2022 *)

a(n)=[-1, 0, 2, 4, 5, 7, 9][n%7+1] + n\7*12 \\ Charles R Greathouse IV, Jul 20 2016

x='x+O('x^99); concat(0, Vec(x^2*(x+1)*(x^5+x^4+x^3+x^2+2)/((x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2))) \\ Jianing Song, Sep 22 2018

G.f.: x^2*(x + 1)*(x^5 + x^4 + x^3 + x^2 + 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 - 70 - 2*(n mod 7) - 2*((n + 1) mod 7) - 2*((n + 2) mod 7) + 5*((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 - 7, a(7k-4) = 12k - 8, 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

A083028 Numbers that are congruent to {0, 2, 3, 5, 7, 8, 11} mod 12.

Original entry on oeis.org

0, 2, 3, 5, 7, 8, 11, 12, 14, 15, 17, 19, 20, 23, 24, 26, 27, 29, 31, 32, 35, 36, 38, 39, 41, 43, 44, 47, 48, 50, 51, 53, 55, 56, 59, 60, 62, 63, 65, 67, 68, 71, 72, 74, 75, 77, 79, 80, 83, 84, 86, 87, 89, 91, 92, 95, 96, 98, 99, 101, 103, 104, 107, 108, 110, 111

Offset: 1

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

The key-numbers of the pitches of a minor scale on a standard chromatic keyboard, with root = 0 and raised seventh.

Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

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: this sequence)
Phrygian mode (E): A083034
Locrian mode (B): A082977
Chords:
Major chord: A083030
Minor chord: A083031
Dominant seventh chord: A083032

[n : n in [0..150] | n mod 12 in [0, 2, 3, 5, 7, 8, 11]]; // Wesley Ivan Hurt, Jul 19 2016

A083028:=n->12*floor(n/7)+[0, 2, 3, 5, 7, 8, 11][(n mod 7)+1]: seq(A083028(n), n=0..100); # Wesley Ivan Hurt, Jul 19 2016

Select[Range[0, 150], MemberQ[{0, 2, 3, 5, 7, 8, 11}, Mod[#, 12]] &] (* Wesley Ivan Hurt, Jul 19 2016 *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 2, 3, 5, 7, 8, 11, 12}, 70] (* Jianing Song, Sep 22 2018 *)

x='x+O('x^99); concat(0, Vec(x^2*(1+x)*(x^5+2*x^4-x^3+3*x^2-x+2)/((x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2))) \\ Jianing Song, Sep 22 2018

G.f.: x^2*(x + 1)*(x^5 + 2*x^4 - x^3 + 3*x^2 - 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 19 2016: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.
a(n) = (84*n - 84 - 9*(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) + 5*((n + 6) mod 7))/49.
a(7k) = 12k - 1, 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)
a(n) = a(n-7) + 12 for n > 7. - Jianing Song, Sep 22 2018

A083032 Numbers that are congruent to {0, 4, 7, 10} mod 12.

Original entry on oeis.org

0, 4, 7, 10, 12, 16, 19, 22, 24, 28, 31, 34, 36, 40, 43, 46, 48, 52, 55, 58, 60, 64, 67, 70, 72, 76, 79, 82, 84, 88, 91, 94, 96, 100, 103, 106, 108, 112, 115, 118, 120, 124, 127, 130, 132, 136, 139, 142, 144, 148, 151, 154, 156, 160, 163, 166, 168, 172

Offset: 1

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

Key-numbers of the pitches of a dominant seventh chord on a standard chromatic keyboard, with root = 0.

G. C. Greubel, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Bisections: A016957, A272975.
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): A083034
Locrian mode (B): A082977
Chords:
Major chord: A083030
Minor chord: A083031
Dominant seventh chord: this sequence

Filtered([0..200],n-> n mod 12=0 or n mod 12=4 or n mod 12=7 or n mod 12=10); # Muniru A Asiru, Sep 22 2018

[(12*n-9+(-1)^n+(-1)^((n+1) div 2)+(-1)^(-(n+1) div 2))/4: n in [1..100]]; // Wesley Ivan Hurt, May 19 2016

A083032:=n->(12*n-9+(-1)^n+(-1)^((n+1)/2)+(-1)^(-(n+1)/2))/4: seq(A083032(n), n=1..100); # Wesley Ivan Hurt, May 19 2016

Select[Range[0,200], MemberQ[{0,4,7,10}, Mod[#,12]]&] (* Harvey P. Dale, Sep 13 2011 *)
LinearRecurrence[{1,0,0,1,-1},{0,4,7,10,12},100] (* G. C. Greubel, Jun 01 2016 *)

my(x='x+O('x^99)); concat(0, Vec(x^2*(4+3*x+3*x^2+2*x^3)/((1+x)*(1+x^2)*(1-x)^2))) \\ Altug Alkan, Sep 21 2018

G.f.: x^2*(4 + 3*x + 3*x^2 + 2*x^3)/((1 + x)*(1 + x^2)*(1 - x)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 19 2016: (Start)
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
a(n) = (12*n - 9 + (-1)^n + (-1)^((n+1)/2) + (-1)^(-(n+1)/2))/4. (End)
a(2k) = A016957(k-1) for k > 0, a(2k-1) = A272975(k). - Wesley Ivan Hurt, Jun 01 2016
E.g.f.: (4 - sin(x) + (6*x - 5)*sinh(x) + (6*x - 4)*cosh(x))/2. - Ilya Gutkovskiy, Jun 01 2016
From Jianing Song, Sep 22 2018: (Start)
a(n) = (12*n - 9 + (-1)^n - 2*sin(n*Pi/2))/4.
a(n) = a(n-4) + 12 for n > 4. (End)
Sum_{n>=2} (-1)^n/a(n) = log(3)/8 - log(2)/12 + sqrt(3)*log(sqrt(3)+2)/12 - (5*sqrt(3)-6)*Pi/72. - Amiram Eldar, Dec 31 2021

A083033 Numbers that are congruent to {0, 2, 3, 5, 7, 9, 10} mod 12.

Original entry on oeis.org

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

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

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.

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).

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

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

[n : n in [0..150] | n mod 12 in [0, 2, 3, 5, 7, 9, 10]]; // Wesley Ivan Hurt, Jul 20 2016

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

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 *)

a(n)=[-2, 0, 2, 3, 5, 7, 9][n%7+1] + n\7*12 \\ Charles R Greathouse IV, Jul 20 2016

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

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

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

Original entry on oeis.org

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

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

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.

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).

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

[n : n in [0..150] | n mod 12 in [0, 2, 4, 6, 7, 9, 11]]; // Wesley Ivan Hurt, Jul 20 2016

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

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 *)

a(n) = 2*(n-1)-2*(n-1)\7; \\ Altug Alkan, Sep 21 2018

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

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

A083030 Numbers that are congruent to {0, 4, 7} mod 12.

Original entry on oeis.org

0, 4, 7, 12, 16, 19, 24, 28, 31, 36, 40, 43, 48, 52, 55, 60, 64, 67, 72, 76, 79, 84, 88, 91, 96, 100, 103, 108, 112, 115, 120, 124, 127, 132, 136, 139, 144, 148, 151, 156, 160, 163, 168, 172, 175, 180, 184, 187, 192, 196, 199, 204, 208, 211, 216, 220

Offset: 1

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

Key-numbers of the pitches of a major common chord on a standard chromatic keyboard, with root = 0.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

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): A083034
Locrian mode (B): A082977
Chords:
Major chord: this sequence
Minor chord: A083031
Dominant seventh chord: A083032

[n : n in [0..300] | n mod 12 in [0, 4, 7]]; // Wesley Ivan Hurt, Jun 14 2016

A083030:=n->4*n-(13+2*cos(2*n*Pi/3))/3: seq(A083030(n), n=1..100); # Wesley Ivan Hurt, Jun 14 2016

Select[Range[0,250], MemberQ[{0,4,7}, Mod[#,12]]&] (* Harvey P. Dale, Apr 17 2014 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 4, 7, 12}, 100] (* Jianing Song, Sep 22 2018 *)

my(x='x+O('x^99)); concat(0, Vec(x^2*(4+3*x+5*x^2)/((1+x+x^2)*(1-x)^2))) \\ Jianing Song, Sep 22 2018

G.f.: x^2*(4 + 3*x + 5*x^2)/((1 + x + x^2)*(1 - x)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jun 14 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 4.
a(n) = 4*n - (13 + 2*cos(2*n*Pi/3))/3.
a(3k) = 12k - 5, a(3k-1) = 12k - 8, a(3k-2) = 12k - 12. (End)
a(n) = a(n-3) + 12 for n > 3. - Jianing Song, Sep 22 2018

A083031 Numbers that are congruent to {0, 3, 7} mod 12.

Original entry on oeis.org

0, 3, 7, 12, 15, 19, 24, 27, 31, 36, 39, 43, 48, 51, 55, 60, 63, 67, 72, 75, 79, 84, 87, 91, 96, 99, 103, 108, 111, 115, 120, 123, 127, 132, 135, 139, 144, 147, 151, 156, 159, 163, 168, 171, 175, 180, 183, 187, 192, 195, 199, 204, 207, 211, 216, 219

Offset: 1

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

Key-numbers of the pitches of a minor common chord on a standard chromatic keyboard, with root = 0.

Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

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): A083034
Locrian mode (B): A082977
Chords:
Major chord: A083030
Minor chord: this sequence
Dominant seventh chord: A083032

[n : n in [0..300] | n mod 12 in [0, 3, 7]]; // Wesley Ivan Hurt, Jun 14 2016

A083031:=n->(12*n-14-cos(2*n*Pi/3)+sqrt(3)*sin(2*n*Pi/3))/3: seq(A083031(n), n=1..100); # Wesley Ivan Hurt, Jun 14 2016

Select[Range[0, 400], MemberQ[{0, 3, 7}, Mod[#, 12]] &] (* Wesley Ivan Hurt, Jun 14 2016 *)
LinearRecurrence[{1, 0, 1, -1}, {0, 3, 7, 12}, 100] (* Jianing Song, Sep 22 2018 *)

x='x+O('x^99); concat(0, Vec(x^2*(3+4*x+5*x^2)/((1+x+x^2)*(1-x)^2))) \\ Jianing Song, Sep 22 2018

G.f.: x^2*(3 + 4*x + 5*x^2)/((1 + x + x^2)*(1 - x)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jun 14 2016: (Start)
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 4.
a(n) = (12*n - 14 - cos(2*n*Pi/3) + sqrt(3)*sin(2*n*Pi/3))/3.
a(3k) = 12k - 5, a(3k-1) = 12k - 9, a(3k-2) = 12k - 12. (End)
a(n) = a(n-3) + 12 for n > 3. - Jianing Song, Sep 22 2018

A083034 Numbers that are congruent to {0, 1, 3, 5, 7, 8, 10} mod 12.

Original entry on oeis.org

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

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

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.

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).

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

[n : n in [0..150] | n mod 12 in [0, 1, 3, 5, 7, 8, 10]]; // Wesley Ivan Hurt, Jul 20 2016

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

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 *)

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

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

cp's OEIS Frontend

A083042 Duplicate of A082977.

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A060107 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.

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A083026 Numbers that are congruent to {0, 2, 4, 5, 7, 9, 11} mod 12.

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A083028 Numbers that are congruent to {0, 2, 3, 5, 7, 8, 11} mod 12.

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A083032 Numbers that are congruent to {0, 4, 7, 10} mod 12.

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A083033 Numbers that are congruent to {0, 2, 3, 5, 7, 9, 10} mod 12.

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

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