A059620 -id:A059620 - cp's OEIS frontend

A209257 A musically inspired Titius-Bode-like sequence based on the geometric division of 4- and 5-dimensional space: Z_(n+1) = 3 * (C(n-1, 0) + C(n-1, 1) + C(n-1, 2) + C(n-1, 3) + C(n-1, 4) + C(n-1, 5)*A059620(n+6)) + 4.

4, 7, 10, 16, 28, 52, 97, 193, 301, 493, 1150, 1162, 3076, 2386, 3283, 10423, 5827, 20659, 9646, 37852, 15112, 18592, 83692, 27331, 133660, 38857, 45832, 251050, 62566, 367318, 83527, 523315, 109375, 124351, 852826, 158872, 1152508, 200140, 223561, 1754809

Offset: 0

Raphie Frank, Jan 14 2013

The classical Titius-Bode version of this sequence is given in A003461.
C(n, 0) + C(n, 1) + C(n, 2) + C(n, 3) + C(n, 4) = A000127(n) = A059173(n+1)/2.
C(n, 0) + C(n, 1) + C(n, 2) + C(n, 3) + C(n, 4) + C(n, 5) = A006261(n) = A059174(n+1)/2.
Where planetary and dwarf-planetary distances from the Sun at semi-major axis are expressed in astronomical units/10, then compare the following (noting that the running correlation coefficient, r, trends upwards as the population size increases):
n = 0, Mercury @ semi-major =   3.8710 vs.   4.0 -->  96.78%.
n = 1, Venus   @ semi-major =   7.2333 vs.   7.0 --> 103.33%.
n = 2, Earth   @ semi-major =  10.0000 vs.  10.0 --> 100.00%,  r = 0.998430.
n = 3, Mars    @ semi-major =  15.2368 vs.  16.0 -->  95.23%,  r = 0.998356.
n = 4, Ceres   @ semi-major =  27.654  vs.  28.0 -->  98.76%,  r = 0.999412.
n = 5, Jupiter @ semi-major =  52.0427 vs.  52.0 --> 100.08%,  r = 0.999809.
n = 6, Saturn  @ semi-major =  95.8202 vs.  97.0 -->  98.78%,  r = 0.999937.
n = 7, Uranus  @ semi-major = 192.2941 vs. 193.0 -->  99.63%,  r = 0.999981.
n = 8, Neptune @ semi-major = 301.0366 vs. 301.0 --> 100.01%,  r = 0.999990.
The correspondence between this sequence and planetary distances breaks down subsequent to Neptune unless one adopts the conceit of considering the outer four dwarf planets -- Pluto, Haumea, MakeMake and Eris -- as one unit occupying one "planetary band" (note that Eris @ perihelion is inside the Kuiper Belt). Then:
n = 9, Pluto/Haumea/MakeMake/Eris @ semi-major ~ 490.492 average vs. 493.0 --> 99.49%, r = 0.999994.
Empirical source: Wikipedia planet pages as of Jan 14 2013.
This sequence originated as part of an attempt to compare and contrast the "good" numerology of Johann Balmer to the "bad" numerology of Titius-Bode. Coincidentally, (Totient(C(31, 0) + C(31, 1) + C(31, 2) + C(31, 3) + C(31, 4)))/10^11 equals 3.6456*10^-7, in meters, the Balmer constant as given by Johann Balmer in 1885.

			Z_1 = 3*((1 - 1 +  1 -  1 +  1) + (-1 * 1)) + 4 =   4,
Z_2 = 3*((1 + 0 +  0 +  0 +  0) +  (0 * 0)) + 4 =   7,
Z_3 = 3*((1 + 1 +  0 +  0 +  0) +  (0 * 0)) + 4 =  10,
Z_4 = 3*((1 + 2 +  1 +  0 +  0) +  (0 * 1)) + 4 =  16,
Z_5 = 3*((1 + 3 +  3 +  1 +  0) +  (0 * 0)) + 4 =  28,
Z_6 = 3*((1 + 4 +  6 +  4 +  1) +  (0 * 1)) + 4 =  52,
Z_7 = 3*((1 + 5 + 10 + 10 +  5) +  (1 * 0)) + 4 =  97,
Z_8 = 3*((1 + 6 + 15 + 20 + 15) +  (6 * 1)) + 4 = 193,
Z_9 = 3*((1 + 7 + 21 + 35 + 35) + (21 * 0)) + 4 = 301.

G. C. Greubel, Table of n, a(n) for n = 0..1000
J. Baez, This Week's Finds in Mathematical Physics (Week 234)
H. Chang, Titius-Bode’s Relation and Distribution of Exoplanets
D. M. F. Chapman, The Titius-Bode Rule, Part 2: Science or Numerology?
Chemteam, The Balmer Formula
A. L. Kholodenko, Role of general relativity and quantum mechanics in dynamics of Solar System
Wikipedia, Dwarf planet
Wikipedia, Planet
Wikipedia, Titius-Bode law

Cf. A003461, A059620, A000127, A059173, A006261, A059174.

[3*(Binomial(n-1,0) + Binomial(n-1,1) + Binomial(n-1,2) + Binomial(n-1,3) + Binomial(n-1,4) + Binomial(n-1,5)*(Floor((5*(n+6) + 7)/12) - Floor((5*(n+6)+2)/12))) + 4: n in [0..30]]; // G. C. Greubel, Jan 07 2018

Z[n_]:= 3*(Binomial[n - 1, 0] + Binomial[n - 1, 1] + Binomial[n - 1, 2] + Binomial[n - 1, 3] + Binomial[n - 1, 4] + Binomial[n - 1, 5]*(Floor[(5 (n + 6) + 7)/12] - Floor[(5 (n + 6) + 2)/12])) + 4; Table[Z[n], {n, 0, 50}] (* G. C. Greubel, Jan 07 2018 *)

{z(n) = 3*(binomial(n-1,0) + binomial(n-1,1) + binomial(n-1,2) + binomial(n-1,3) + binomial(n-1,4) + binomial(n-1,5)*(floor((5*(n+6) + 7)/12) - floor((5*(n+6)+2)/12))) + 4};
for(n=0,30, print1(z(n), ", ")) \\ G. C. Greubel, Jan 07 2018

Z_(n+1) = 3 * (C(n-1, 0) + C(n-1, 1) + C(n-1, 2) + C(n-1, 3) + C(n-1, 4) + C(n-1, 5)*(floor((5*(n+6)+7)/12) - floor((5*(n+6)+2)/12))) + 4.

a(18) corrected by G. C. Greubel, Jan 07 2018

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.

Original entry on oeis.org

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

A060106 Numbers that are congruent to {1, 4, 6, 9, 11} mod 12. The ebony keys on a piano, starting with A0 = the 0th key.

Original entry on oeis.org

1, 4, 6, 9, 11, 13, 16, 18, 21, 23, 25, 28, 30, 33, 35, 37, 40, 42, 45, 47, 49, 52, 54, 57, 59, 61, 64, 66, 69, 71, 73, 76, 78, 81, 83, 85, 88, 90, 93, 95, 97, 100, 102, 105, 107, 109, 112, 114, 117, 119, 121, 124, 126, 129, 131, 133, 136, 138, 141, 143, 145, 148

Offset: 1

Henry Bottomley, Feb 27 2001

A piano sequence since if a(n) < 88 then A059620(a(n)) = 1.

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

Cf. A059620, A081032. Complement of A060107.

Vec(x*(1 + 3*x + 2*x^2 + 3*x^3 + 2*x^4 + x^5) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ Colin Barker, Oct 14 2019

a(n) = a(n-5) + 12.
a(n) = A081032(n) - 1 for 1 <= n <= 36. - Jianing Song, Oct 14 2019
From Colin Barker, Oct 14 2019: (Start)
G.f.: x*(1 + 3*x + 2*x^2 + 3*x^3 + 2*x^4 + x^5) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6) for n > 6.
(End)

A214832 Integer part of A440 piano key frequencies, start with A0 = the 1st key.

Original entry on oeis.org

27, 29, 30, 32, 34, 36, 38, 41, 43, 46, 48, 51, 55, 58, 61, 65, 69, 73, 77, 82, 87, 92, 97, 103, 110, 116, 123, 130, 138, 146, 155, 164, 174, 184, 195, 207, 220, 233, 246, 261, 277, 293, 311, 329, 349, 369, 391, 415, 440, 466, 493, 523, 554, 587, 622, 659, 698, 739, 783, 830, 880, 932, 987, 1046, 1108, 1174, 1244, 1318, 1396, 1479, 1567, 1661, 1760, 1864, 1975, 2093, 2217, 2349, 2489, 2637, 2793, 2959, 3135, 3322, 3520, 3729, 3951, 4186

Offset: 1

Jon Perry, Mar 07 2013

A254531(a(k)) = k, k = 1..88. - Reinhard Zumkeller, Feb 04 2015

			Middle C is 261.626 Hz so a(40) = 261.

BGfL, A Virtual Keyboard
Wikipedia, Piano Key Frequencies
Wikipedia, Twelfth root of two
Index entries for sequences based on music

Cf. A059620, A079731, A131062, A101285, A131071.

Cf. A254531, A010774.

a214832 = floor . (* 440) . (2 **) . (/ 12) . fromIntegral . subtract 49
-- Reinhard Zumkeller, Nov 23 2014

for (i=1;i<=88;i++) document.write(Math.floor(Math.pow(2,(i-49)/12)*440)+", ");

PARI
```
a(n)=floor(440*2^((n-49)/12));
```

a(n) = floor[2^((n-49)/12)*440] (Hz) for 1 <= n <= 88.

A081031 Positions of white keys on piano keyboard, starting with A0 = the 1st key.

Original entry on oeis.org

1, 3, 4, 6, 8, 9, 11, 13, 15, 16, 18, 20, 21, 23, 25, 27, 28, 30, 32, 33, 35, 37, 39, 40, 42, 44, 45, 47, 49, 51, 52, 54, 56, 57, 59, 61, 63, 64, 66, 68, 69, 71, 73, 75, 76, 78, 80, 81, 83, 85, 87, 88

Offset: 1

David W. Wilson, Mar 02 2003

			First, 3rd, 4th, 6th, etc. keys of piano keyboard are white.

Robbert Fokkink, The Pell Tower and Ostronometry, arXiv:2309.01644 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A059620, A081032, A060107.

Join[{1,3},Flatten[Table[12n+{4,6,8,9,11,13,15},{n,0,6}]],{88}] (* Harvey P. Dale, Mar 15 2013 *)
LinearRecurrence[{1,0,0,0,0,0,1,-1},{1,3,4,6,8,9,11,13},52] (* Harvey P. Dale, May 14 2023 *)

a(n) = floor((12n-3)/7).
From Chai Wah Wu, Sep 11 2018: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.
G.f. for a keyboard with infinite number of keys: x*(x^7 + 2*x^6 + x^5 + 2*x^4 + 2*x^3 + x^2 + 2*x + 1)/(x^8 - x^7 - x + 1). (End)
a(n) = A060107(n) + 1 for 1 <= n <= 36. - Jianing Song, Oct 14 2019

A081032 Positions of black keys on piano keyboard, starting with A0 = the 1st key.

Original entry on oeis.org

2, 5, 7, 10, 12, 14, 17, 19, 22, 24, 26, 29, 31, 34, 36, 38, 41, 43, 46, 48, 50, 53, 55, 58, 60, 62, 65, 67, 70, 72, 74, 77, 79, 82, 84, 86

Offset: 1

David W. Wilson, Mar 02 2003

			2nd, 5th, 7th, 10th, etc. keys of piano keyboard are black.

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

Cf. A059620, A081031, A060106.

LinearRecurrence[{1,0,0,0,1,-1},{2,5,7,10,12,14},36] (* Harvey P. Dale, Sep 15 2018 *)

a(n) = floor((12n+2)/5).
From Chai Wah Wu, Sep 11 2018: (Start)
a(n) = a(n-1) + a(n-5) - a(n-6) for n > 6.
G.f. for a keyboard with an infinite number of keys: x*(2*x^4 + 3*x^3 + 2*x^2 + 3*x + 2)/(x^6 - x^5 - x + 1). (End)
a(n) = A060106(n) + 1 for 1 <= n <= 36. - Jianing Song, Oct 14 2019

A356464 Number of black keys in each group of black keys on a standard 88-key piano (left to right).

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3

Offset: 1

Peter Woodward, Aug 08 2022

On a standard piano keyboard, the black keys appear in groups of two and three, with each group separated from adjacent groups by the presence of two white keys that have no black key between them.
The black keys in a group of two are C#/Db and D#/Eb; the black keys in a group of three are F#/Gb, G#/Ab, and A#/Bb.
The A#/Bb key near the left end of the keyboard is a special case; it is the only black key in its group because the white A key to its left is the leftmost key on the keyboard.

			From _Jon E. Schoenfield_, Aug 12 2022: (Start)
In the diagram below, five octaves (i.e., sets of 12 consecutive keys) have been omitted (as represented by the ellipses):
.
    n |  1       2         3       ...     14        15
  ----+---------------------------------------------------------
  a(n)|  1       2         3       ...      2         3
    ______________________________ ... _________________________
      | |/| | |/||/| | |/||/||/| |     | |/||/| | |/||/||/| |  |
      | |/| | |/||/| | |/||/||/| |     | |/||/| | |/||/||/| |  |
      | |/| | |/||/| | |/||/||/| |     | |/||/| | |/||/||/| |  |
      | |_| | |_||_| | |_||_||_| |     | |_||_| | |_||_||_| |  |
      |  |  |  |  |  |  |  |  |  |     |  |  |  |  |  |  |  |  |
      |  |  |  |  |  |  |  |  |  |     |  |  |  |  |  |  |  |  |
      |__|__|__|__|__|__|__|__|__|     |__|__|__|__|__|__|__|__|
       A  B  C  D  E  F  G  A  B   ...  C  D  E  F  G  A  B  C
(End)

Cf. A059620, A060107, A060106, A081031, A081032.

Cf. A329207.

cp's OEIS Frontend

A209257 A musically inspired Titius-Bode-like sequence based on the geometric division of 4- and 5-dimensional space: Z_(n+1) = 3 * (C(n-1, 0) + C(n-1, 1) + C(n-1, 2) + C(n-1, 3) + C(n-1, 4) + C(n-1, 5)*A059620(n+6)) + 4.

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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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A060106 Numbers that are congruent to {1, 4, 6, 9, 11} mod 12. The ebony keys on a piano, starting with A0 = the 0th key.

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A214832 Integer part of A440 piano key frequencies, start with A0 = the 1st key.

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A081031 Positions of white keys on piano keyboard, starting with A0 = the 1st key.

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A081032 Positions of black keys on piano keyboard, starting with A0 = the 1st key.

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A356464 Number of black keys in each group of black keys on a standard 88-key piano (left to right).

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