A083026 -id:A083026 - cp's OEIS frontend

A319451 Numbers that are congruent to {0, 3, 6} mod 12; a(n) = 3floor(4n/3).

0, 3, 6, 12, 15, 18, 24, 27, 30, 36, 39, 42, 48, 51, 54, 60, 63, 66, 72, 75, 78, 84, 87, 90, 96, 99, 102, 108, 111, 114, 120, 123, 126, 132, 135, 138, 144, 147, 150, 156, 159, 162, 168, 171, 174, 180, 183, 186, 192, 195, 198, 204, 207, 210, 216, 219, 222, 228

Offset: 0

Jianing Song, Sep 19 2018

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

Jianing Song, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A004773, A047464.
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
Third chords:
Major chord (F,C,G): A083030
Minor chord (D,A,E): A083031
Diminished chord (B): this sequence
Seventh chords:
Major seventh chord (F,C): A319280
Dominant seventh chord (G): A083032
Minor seventh chord (D,A,E): A319279
Half-diminished seventh chord (B): A319452

Filtered([0..230],n->n mod 12 = 0 or n mod 12 = 3 or n mod 12 = 6); # Muniru A Asiru, Oct 24 2018

[n : n in [0..150] | n mod 12 in [0, 3, 6]]

seq(3*floor(4*n/3),n=0..60); # Muniru A Asiru, Oct 24 2018

Select[Range[0, 200], MemberQ[{0, 3, 6}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 1, -1}, {0, 3, 6, 12}, 100]
Table[4n-1+Sin[Pi/3(2n+1)]/Sin[Pi/3],{n,0,99}] (* Federico Provvedi, Oct 23 2018 *)

PARI
```
a(n)=3*(4*n\3)
```

for n in range(0,60): print(3*int(4*n/3), end=", ") # Stefano Spezia, Dec 07 2018

a(n) = a(n-3) + 12 for n > 2.
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 3.
G.f.: 3*(1 + x + 2*x^2)/((1 - x)*(1 - x^3)).
a(n) = 3*A004773(n) = 3*(floor(n/3) + n).
a(n) = 4*n - 1 + sin((Pi/3)*(2*n + 1))/sin(Pi/3). - Federico Provvedi, Oct 23 2018
E.g.f.: (3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/(3*exp(x/2)) - exp(x)*(1 - 4*x). - Franck Maminirina Ramaharo, Nov 27 2018
Sum_{n>=2} (-1)^n/a(n) = (sqrt(2)-1)*Pi/24 + (2-sqrt(2))*log(2)/24 + sqrt(2)*log(2+sqrt(2))/12. - Amiram Eldar, Dec 30 2021

A319279 Numbers that are congruent to {0, 3, 7, 10} mod 12.

Original entry on oeis.org

0, 3, 7, 10, 12, 15, 19, 22, 24, 27, 31, 34, 36, 39, 43, 46, 48, 51, 55, 58, 60, 63, 67, 70, 72, 75, 79, 82, 84, 87, 91, 94, 96, 99, 103, 106, 108, 111, 115, 118, 120, 123, 127, 130, 132, 135, 139, 142, 144, 147, 151, 154, 156, 159, 163, 166, 168, 171, 175, 178

Offset: 1

Jianing Song, Sep 16 2018

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

Apart from the offset the same as A013574. - R. J. Mathar, Sep 27 2018

Jianing Song, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-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
Third chords:
Major chord (F,C,G): A083030
Minor chord (D,A,E): A083031
Diminished chord (B): A319451
Seventh chords:
Major seventh chord (F,C): A319280
Dominant seventh chord (G): A083032
Minor seventh chord (D,A,E): this sequence
Half-diminished seventh chord (B): A319452

[n : n in [0..150] | n mod 12 in [0, 3, 7, 10]]

Select[Range[0, 200], MemberQ[{0, 3, 7, 10}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 3, 7, 10, 12}, 100]

x='x+O('x^99); concat(0, Vec(x^2*(3+x+2*x^2)/((x^2+1)*(x-1)^2)))

a(n) = a(n-4) + 12 for n > 4.
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
G.f.: x^2*(3 + x + 2*x^2)/((x^2 + 1)*(x - 1)^2).
a(n) = (6*n - 5 + sqrt(2)*cos(Pi*n/2 + Pi/4))/2.
E.g.f.: ((6x - 5)*e^x + sqrt(2)*cos(x + Pi/4) + 4)/2.

A319280 Numbers that are congruent to {0, 4, 7, 11} mod 12.

Original entry on oeis.org

0, 4, 7, 11, 12, 16, 19, 23, 24, 28, 31, 35, 36, 40, 43, 47, 48, 52, 55, 59, 60, 64, 67, 71, 72, 76, 79, 83, 84, 88, 91, 95, 96, 100, 103, 107, 108, 112, 115, 119, 120, 124, 127, 131, 132, 136, 139, 143, 144, 148, 151, 155, 156, 160, 163, 167, 168, 172, 175, 179

Offset: 1

Jianing Song, Sep 16 2018

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

Jianing Song, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-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): A083034
Locrian mode (B): A082977
Third chords:
Major chord (F,C,G): A083030
Minor chord (D,A,E): A083031
Diminished chord (B): A319451
Seventh chords:
Major seventh chord (F,C): this sequence
Dominant seventh chord (G): A083032
Minor seventh chord (D,A,E): A319279
Half-diminished seventh chord (B): A319452

[n : n in [0..150] | n mod 12 in [0, 4, 7, 11]];

Select[Range[0, 200], MemberQ[{0, 4, 7, 11}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 4, 7, 11, 12}, 100]

my(x='x+O('x^99)); concat(0, Vec(x^2*(4+3*x+4*x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2)))

a(n) = a(n-4) + 12 for n > 4.
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
G.f.: x^2*(4 + 3*x + 4*x^2 + x^3)/((1 + x)*(1 + x^2)*(1 - x)^2).
a(n) = (6*n - 4 + (-1)^n + sqrt(2)*cos(Pi*n/2 + Pi/4))/2.
E.g.f.: ((6*x - 3)*cosh(x) + (6*x - 5)*sinh(x) + sqrt(2)*cos(x + Pi/4) + 2)/2.
Sum_{n>=2} (-1)^n/a(n) = log(3)/8 + log(2+sqrt(3))/(2*sqrt(3)) - 5*sqrt(3)*Pi/72. - Amiram Eldar, Dec 30 2021

A319452 Numbers that are congruent to {0, 3, 6, 10} mod 12.

Original entry on oeis.org

0, 3, 6, 10, 12, 15, 18, 22, 24, 27, 30, 34, 36, 39, 42, 46, 48, 51, 54, 58, 60, 63, 66, 70, 72, 75, 78, 82, 84, 87, 90, 94, 96, 99, 102, 106, 108, 111, 114, 118, 120, 123, 126, 130, 132, 135, 138, 142, 144, 147, 150, 154, 156, 159, 162, 166, 168, 171, 174, 178

Offset: 1

Jianing Song, Sep 19 2018

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

Jianing Song, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-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): A083034
Locrian mode (B): A082977
Third chords:
Major chord (F,C,G): A083030
Minor chord (D,A,E): A083031
Diminished chord (B): A319451
Seventh chords:
Major seventh chord (F,C): A319280
Dominant seventh chord (G): A083032
Minor seventh chord (D,A,E): A319279
Half-diminished seventh chord (B): this sequence

[n : n in [0..150] | n mod 12 in [0, 3, 6, 10]]

Select[Range[0, 200], MemberQ[{0, 3, 6, 10}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 3, 6, 10, 12}, 100]

my(x='x+O('x^99)); concat(0, Vec(x^2*(3+3*x+4*x^2+2*x^3)/((1+x)*(1+x^2)*(1-x)^2)))

a(n) = a(n-4) + 12 for n > 4.
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
G.f.: x^2*(3 + 3*x + 4*x^2 + 2*x^3)/((1 + x)*(1 + x^2)*(1 - x)^2).
a(n) = (12*n - 11 + (-1)^n + 2*cos(Pi*n/2))/4.
E.g.f.: ((6*x - 5)*cosh(x) + (6*x - 6)*sinh(x) + cos(x) + 4)/2.
Sum_{n>=2} (-1)^n/a(n) = log(12)/8 - (sqrt(3)-1)*Pi/24. - Amiram Eldar, Dec 30 2021

A291454 Number of half tones between successive pitches in a major scale.

Original entry on oeis.org

2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 1

Offset: 1

Halfdan Skjerning, Aug 24 2017

In music theory the repeating sequence '2,2,1,2,2,2,1' is the number of steps of half tones in pitch between the tones of a major scale. Starting at, for example, the tone 'C' that is the first tone of the C major scale, 2 half tones up leads to 'D', which is the second tone in the scale. The scale then is: C,D,E,F,G,A,B and C. Starting at another term in the sequence will produce a different scale; for example, '2,1,2,2,1,2,2' will produce a minor scale.
From Robert G. Wilson v, Aug 25 2017: (Start)
First forward difference of A083026.
Decimal expansion of 737407/3333333. (End)

Muniru A Asiru, Table of n, a(n) for n = 1..1000
Wikipedia, Scale (music)

Cf. A083026, A286748.

[12*(n+1) div 7 - 12*n div 7:  n in [1..80]]; // Vincenzo Librandi, Oct 21 2018

a:=proc(n) floor(12*(n+1)/7-floor(12*n/7)) end: seq(a(n),n=1..110); # Muniru A Asiru, Oct 19 2018

Table[{2, 2, 1, 2, 2, 2, 1}, 15] // Flatten  (* Robert G. Wilson v, Aug 25 2017 *)
Table[Floor[12/7 (k + 1)] - Floor[12/7 k], {k, 1, 100}] (* Federico Provvedi,Oct 18 2018 *)

a(n)=[1,2,2,1,2,2,2][n%7+1] \\ Charles R Greathouse IV, Aug 26 2017

a(n) = floor(12*(n+1)/7) - floor(12*n/7). - Federico Provvedi, Oct 18 2018

Dirichlet g.f.: 2*zeta(s) - 7^(-s)*(zeta(s,3/7) + zeta(s)). - Federico Provvedi, Aug 27 2021

A175884 Numbers that are congruent to {0, 2, 4, 7, 9} mod 12.

Original entry on oeis.org

0, 2, 4, 7, 9, 12, 14, 16, 19, 21, 24, 26, 28, 31, 33, 36, 38, 40, 43, 45, 48, 50, 52, 55, 57, 60, 62, 64, 67, 69, 72, 74, 76, 79, 81, 84, 86, 88, 91, 93, 96, 98, 100, 103, 105, 108, 110, 112, 115, 117, 120, 122, 124, 127, 129, 132, 134, 136, 139, 141, 144, 146, 148, 151

Offset: 1

Bill Shillito (DMAshura(AT)gmail.com), Oct 08 2010

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

The pentatonic scale can also be obtained by omitting the 4th and 7th notes from the diatonic scale, so a(n) = A083026(A032796(n)). - Federico Provvedi, Sep 10 2022

G. C. Greubel, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)
Wikipedia, Major pentatonic scale
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A005843, A057354, A002266.

Subset of A083026 with exact index A032796.

Filtered([0..151],n->n mod 12 = 0 or n mod 12 = 2 or n mod 12 = 4 or n mod 12 = 7 or n mod 12 = 9); # Muniru A Asiru, Oct 24 2018

[Floor(12*(n-1)/5): n in [1..100]]; // G. C. Greubel, Oct 23 2018

seq(floor(12*(n-1)/5),n=1..65); # Muniru A Asiru, Oct 24 2018

fQ[n_] := MemberQ[{0, 2, 4, 7, 9}, Mod[n, 12]]; Select[ Range[0, 152], fQ] (* Robert G. Wilson v, Oct 09 2010 *)
Table[2n-1+Floor[(n-4)/5]+Floor[(n-1)/5],{n, 100}] (* Federico Provvedi, Jan 13 2018 *)
Quotient[12(Range[100]-1), 5] (* Federico Provvedi, Oct 19 2018 *)

a(n)=([0,1,0,0,0,0; 0,0,1,0,0,0; 0,0,0,1,0,0; 0,0,0,0,1,0; 0,0,0,0,0,1; -1,1,0,0,0,1]^n*[0;2;4;7;9;12])[1,1] \\ for offset 0; Charles R Greathouse IV, Jul 06 2017

vector(100, n, floor(12*(n-1)/5)) \\ G. C. Greubel, Oct 23 2018

G.f.: x^2*(2 + 2*x + 3*x^2 + 2*x^3 + 3*x^4) / ((x^4 + x^3 + x^2 + x + 1)*(x-1)^2). - R. J. Mathar, Jul 10 2015
a(n) = 2*n - 1 + floor((n-4)/5) + floor((n-1)/5). - Federico Provvedi, Jan 13 2018
a(n) = floor(12*(n-1)/5). - Federico Provvedi, Oct 19 2018
a(n) = A005843(n) + A057354(n). - Federico Provvedi, Sep 10 2022

Offset change by G. C. Greubel, Oct 23 2018

A371902 Positive integers whose binary form follows the periodic pattern 1101110: the concatenation of halftones 2 2 1 2 2 2 1, diminished by one, between successive pitches in the Ionian Major Scale.

Original entry on oeis.org

1, 3, 6, 13, 27, 55, 110, 221, 443, 886, 1773, 3547, 7095, 14190, 28381, 56763, 113526, 227053, 454107, 908215, 1816430, 3632861, 7265723, 14531446, 29062893, 58125787, 116251575, 232503150, 465006301, 930012603, 1860025206, 3720050413

Offset: 1

Federico Provvedi, Apr 13 2024

The periodic binary digits of 55/107 is the pattern sequence A291454(n)-1 which is the new bit introduced into a(n): a(n+1) = 2*a(n) + A291454(n) - 1.

			For n=10, playing 10 + 1 = 11 notes of the major scale (in Ionian mode), the 10 intervals between the pitches C D E F G A B C' D' E' F' expressed in halftones are 2, 2, 1, 2, 2, 2, 1, 2, 2, 1, whose values diminished by one give the binary form '1101110110', which in decimal is 886, hence a(10) = 886.

Paolo Xausa, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,0,0,0,0,0,1,-2).

Cf. A083026, A291454, A000225, A234046

Floor[110/127*2^Range[50]] (* Paolo Xausa, Jun 21 2024 *)

a(n) = floor((110/127)*2^n).
D.g.f.: z^2*(z^5 + z^4 + z^2 + z + 1)/((2 - z) (1 - z^7)) = z * Dgf(A000225) * Dgf(A234046).
G.f.: x*(1 + x + x^3 + x^4 + x^5)/((1 - 2*x)*(1 - x^7)). - Stefano Spezia, May 04 2024

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A319451 Numbers that are congruent to {0, 3, 6} mod 12; a(n) = 3*floor(4*n/3).

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A319279 Numbers that are congruent to {0, 3, 7, 10} mod 12.

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A319280 Numbers that are congruent to {0, 4, 7, 11} mod 12.

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A319452 Numbers that are congruent to {0, 3, 6, 10} mod 12.

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A291454 Number of half tones between successive pitches in a major scale.

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A175884 Numbers that are congruent to {0, 2, 4, 7, 9} mod 12.

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A319451 Numbers that are congruent to {0, 3, 6} mod 12; a(n) = 3floor(4n/3).