A082977 -id:A082977 - cp's OEIS frontend

A083120 Numbers that are congruent to {0, 2, 4, 5, 7, 9, 10} mod 12.

0, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, 19, 21, 22, 24, 26, 28, 29, 31, 33, 34, 36, 38, 40, 41, 43, 45, 46, 48, 50, 52, 53, 55, 57, 58, 60, 62, 64, 65, 67, 69, 70, 72, 74, 76, 77, 79, 81, 82, 84, 86, 88, 89, 91, 93, 94, 96, 98, 100, 101, 103, 105, 106, 108, 110

Offset: 1

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

Key-numbers of the pitches of a Mixolydian mode scale on a standard chromatic keyboard, with root = 0. A Mixolydian mode scale can, for example, be played on consecutive white keys of a standard keyboard, starting on the root tone G.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
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): this sequence
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, 10]]; // Wesley Ivan Hurt, Jul 20 2016

A083120:= n-> 12*floor((n-1)/7)+[0, 2, 4, 5, 7, 9, 10][((n-1) mod 7)+1]:
seq(A083120(n), n=1..100); # Wesley Ivan Hurt, Jul 20 2016

Select[Range[0,120], MemberQ[{0,2,4,5,7,9,10}, Mod[#,12]]&] (* Harvey P. Dale, Feb 20 2011 *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 2, 4, 5, 7, 9, 10, 12}, 70] (* Jianing Song, Sep 22 2018 *)
Quotient[4 (3 # - 2), 7] & /@ Range[96] (* Federico Provvedi, Nov 06 2023 *)

a(n)=[-2, 0, 2, 4, 5, 7, 9][n%7+1] + n\7*12 \\ Charles R Greathouse IV, Jul 21 2016

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

G.f.: x^2*(2 + 2*x + x^2 + 2*x^3 + 2*x^4 + 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) = a(n-1) + a(n-7) - a(n-8) for n > 8.
a(n) = (84*n - 77 + 5*(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) - 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 - 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
a(n) = floor(4 * (3*n - 2) / 7). Federico Provvedi, Nov 06 2023

A000210 A Beatty sequence: floor(n*(e-1)).

Original entry on oeis.org

1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 18, 20, 22, 24, 25, 27, 29, 30, 32, 34, 36, 37, 39, 41, 42, 44, 46, 48, 49, 51, 53, 54, 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, 113, 115, 116

Offset: 1

N. J. A. Sloane

nonn

The first 38 terms coincide with the corresponding terms of A082977, i.e., numbers that are congruent to {0, 1, 3, 5, 6, 8, 10} mod 12. - Giovanni Resta, Mar 24 2006

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
I. G. Connell, Some properties of Beatty sequences II, Canad. Math. Bull., 3 (1960), 17-22.
Index entries for sequences related to Beatty sequences

a:= n-> floor (n*(exp(1)-1)): seq (a(n), n=1..200); # Alois P. Heinz, Aug 25 2008

Table[Floor[n*(E - 1)], {n, 0, 100}] (* T. D. Noe, Jan 21 2013 *)

More terms from James Sellers, Jul 06 2000

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

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

A047329 Numbers that are congruent to {1, 3, 5, 6} mod 7.

Original entry on oeis.org

1, 3, 5, 6, 8, 10, 12, 13, 15, 17, 19, 20, 22, 24, 26, 27, 29, 31, 33, 34, 36, 38, 40, 41, 43, 45, 47, 48, 50, 52, 54, 55, 57, 59, 61, 62, 64, 66, 68, 69, 71, 73, 75, 76, 78, 80, 82, 83, 85, 87, 89, 90, 92, 94, 96, 97, 99, 101, 103, 104, 106, 108, 110, 111

Offset: 1

N. J. A. Sloane

Robert Fludd, Utriusque Cosmi ... Historia, Oppenheim, 1617-1619.

Robert Fludd, Page 158 of "Utriusque Cosmi" in Beinecke Rare Book and Manuscript Library Photonegatives Collection.
Robert Fludd, Larger version of the same image
Robert Fludd, Utriusque Cosmi, Maioris scilicet et Minoris, metaphysica, physica, atque technica Historia, available as ZIP or PDF download.
Wikipedia, Robert Fludd
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

Cf. A047280, A047383, A082977.

a047329 n = a047329_list !! (n-1)
a047329_list = [1, 3, 5, 6] ++ map (+ 7) a047329_list
-- Reinhard Zumkeller, Jan 07 2014

[Floor((7*n-1)/4) : n in [1..100]]; // Wesley Ivan Hurt, May 21 2016

A047329:=n->floor((7*n-1)/4): seq(A047329(n), n=1..100); # Wesley Ivan Hurt, May 21 2016

Table[Floor[(7*n-1)/4], {n, 80}] (* Wesley Ivan Hurt, May 21 2016 *)
#+{1,3,5,6}&/@(7*Range[0,20])//Flatten (* Harvey P. Dale, Jan 07 2021 *)

a(n) = floor((7n-1)/4). - Gary Detlefs, Mar 07 2010
G.f.: (x*(1+2*x+2*x^2+x^3+x^4)) / ((1+x)*(x^2+1)*(x-1)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, May 21 2016: (Start)
a(n) = a(n-1)+a(n-4)-a(n-5) for n>5.
a(n) = (14n-5-i^(2n)-(1+i)*i^(-n)-(1-i)*i^n)/8 where i=sqrt(-1).
a(2n) = A047280(n), a(2n-1) = A047383(n). (End)
E.g.f.: (4 - sin(x) - cos(x) + (7*x - 2)*sinh(x) + (7*x - 3)*cosh(x))/4. - Ilya Gutkovskiy, May 21 2016

Fludd reference from Brendan McKay, May 27 2003

More terms from Wesley Ivan Hurt, May 21 2016

cp's OEIS Frontend

A083120 Numbers that are congruent to {0, 2, 4, 5, 7, 9, 10} mod 12.

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A000210 A Beatty sequence: floor(n*(e-1)).

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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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A047329 Numbers that are congruent to {1, 3, 5, 6} mod 7.

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