cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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

Views

Author

Jianing Song, Sep 19 2018

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • GAP
    Filtered([0..230],n->n mod 12 = 0 or n mod 12 = 3 or n mod 12 = 6); # Muniru A Asiru, Oct 24 2018
  • Magma
    [n : n in [0..150] | n mod 12 in [0, 3, 6]]
  • Maple
    seq(3*floor(4*n/3),n=0..60); # Muniru A Asiru, Oct 24 2018
  • Mathematica
    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)
  • Python
    for n in range(0,60): print(3*int(4*n/3), end=", ") # Stefano Spezia, Dec 07 2018

Formula

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

Views

Author

Jianing Song, Sep 16 2018

Keywords

Comments

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

Links

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): 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

Programs

  • Magma
    [n : n in [0..150] | n mod 12 in [0, 3, 7, 10]]
  • Mathematica
    Select[Range[0, 200], MemberQ[{0, 3, 7, 10}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 3, 7, 10, 12}, 100]
  • PARI
    x='x+O('x^99); concat(0, Vec(x^2*(3+x+2*x^2)/((x^2+1)*(x-1)^2)))

Formula

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.

A190785 Numbers that are congruent to {0, 2, 3, 5, 7, 9, 11} mod 12.

Original entry on oeis.org

0, 2, 3, 5, 7, 9, 11, 12, 14, 15, 17, 19, 21, 23, 24, 26, 27, 29, 31, 33, 35, 36, 38, 39, 41, 43, 45, 47, 48, 50, 51, 53, 55, 57, 59, 60, 62, 63, 65, 67, 69, 71, 72, 74, 75, 77, 79, 81, 83, 84, 86, 87, 89, 91, 93, 95, 96, 98, 99, 101, 103, 105, 107, 108, 110
Offset: 1

Views

Author

Roberto Bertocco, May 26 2011

Keywords

Comments

The key-numbers of the pitches of a ascending melodic minor scale on a standard chromatic keyboard, with root = 0 and raised seventh.
First differences are period 7: repeat [1,2,2,2,2,1,2]. - Bruno Berselli, May 27 2011

Links

Crossrefs

Cf. A083028.

Programs

  • Magma
    [n: n in [0..110] | n mod 12 in [0, 2, 3, 5, 7, 9, 11]]; // Bruno Berselli, May 27 2011
  • Maple
    A190785:=n->12*floor(n/7)+[0, 2, 3, 5, 7, 9, 11][(n mod 7)+1]: seq(A190785(n), n=0..100); # Wesley Ivan Hurt, Jul 21 2016
  • Mathematica
    Union[Flatten[Table[12n + {0, 2, 3, 5, 7, 9, 11}, {n, 0, 8}]]] (* Alonso del Arte, Jun 11 2011 *)
  • PARI
    a(n)=n\7*12+[0,2,3,5,7,9,11][n%7+1] \\ Charles R Greathouse IV, Jun 08 2011
  • Python
    def A190785(n):
    a, b = divmod(n-1,7)
    return (0,2,3,5,7,9,11)[b]+12*a # Chai Wah Wu, Jan 26 2023

Formula

a(n) = a(n-1) + a(n-7) - a(n-8) for n>8; G.f.: ( 2+x+2*x^2+2*x^3+2*x^4+2*x^5+x^6 ) / ( (x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2 ). - R. J. Mathar, May 26 2011
a(n) = 2*n-floor(2*n/7)-floor(((n-4) mod 7)/5). - Rolf Pleisch, Jun 11 2011
From Wesley Ivan Hurt, Jul 21 2016: (Start)
a(n) = a(n-7) + 12 for n>7.
a(n) = (84*n - 77 - 2*(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) + 5*((n+6) mod 7))/49.
a(7*k) = 12*k-1, a(7*k-1) = 12*k-3, a(7*k-2) = 12*k-5, a(7*k-3) = 12*k-7, a(7*k-4) = 12*k-9, a(7*k-5) = 12*k-10, a(7*k-6) = 12*k-12. (End)

Extensions

Zero prepended by Wesley Ivan Hurt, Jul 21 2016

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

Views

Author

Jianing Song, Sep 16 2018

Keywords

Comments

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

Links

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): 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

Programs

  • Magma
    [n : n in [0..150] | n mod 12 in [0, 4, 7, 11]];
  • Mathematica
    Select[Range[0, 200], MemberQ[{0, 4, 7, 11}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 4, 7, 11, 12}, 100]
  • PARI
    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)))

Formula

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

Views

Author

Jianing Song, Sep 19 2018

Keywords

Comments

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

Links

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): 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

Programs

  • Magma
    [n : n in [0..150] | n mod 12 in [0, 3, 6, 10]]
  • Mathematica
    Select[Range[0, 200], MemberQ[{0, 3, 6, 10}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 3, 6, 10, 12}, 100]
  • PARI
    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)))

Formula

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

A349707 Numbers that are congruent to {0, 1, 4, 6, 8, 10, 11} (mod 12).

Original entry on oeis.org

0, 1, 4, 6, 8, 10, 11, 12, 13, 16, 18, 20, 22, 23, 24, 25, 28, 30, 32, 34, 35, 36, 37, 40, 42, 44, 46, 47, 48, 49, 52, 54, 56, 58, 59, 60, 61, 64, 66, 68, 70, 71, 72, 73, 76, 78, 80, 82, 83, 84, 85, 88, 90, 92, 94, 95, 96, 97, 100, 102, 104, 106, 107, 108, 109
Offset: 1

Views

Author

Roberto Bertocco, Nov 26 2021

Keywords

Comments

Terms are the key numbers of the pitches of an Enigmatic scale on a standard chromatic keyboard, with root = 0.

Links

Crossrefs

Cf. A083028.

Programs

  • Mathematica
    upto=200;Select[Range[0,upto],MemberQ[{0,1,4,6,8,10,11},Mod[#,12]]&] (* Paolo Xausa, Nov 30 2021 *)
nterms=100;LinearRecurrence[{1,0,0,0,0,0,1,-1},{0,1,4,6,8,10,11,12},nterms] (* Paolo Xausa, Nov 30 2021 *)
  • Python
    def a(n): return 12*((n-1)//7) + [0, 1, 4, 6, 8, 10, 11][(n-1)%7]
print([a(n) for n in range(1, 66)]) # Michael S. Branicky, Dec 02 2021

Formula

G.f.: x^2*(1 + 3*x + 2*x^2 + 2*x^3 + 2*x^4 + x^5 + x^6)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)). - Stefano Spezia, Dec 01 2021
a(n) = a(n-7) + 12 for n >= 8. - Michael S. Branicky, Dec 02 2021
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