A081032 -id:A081032 - cp's OEIS frontend

A059620 Colors of the 88 keys of the standard piano: white keys = 0, black keys = 1, start with A0 = the 0th key.

0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0

Offset: 0

Naohiro Nomoto, Feb 19 2001

			.1..1.1..1.1.1..1.1..1.1.1..1.1..1.1.1..1.1..1.1.1..1.1..1.1.1..1.1..1.1.1..1.1..1.1.1
0.00.0.00.0.0.00.0.00.0.0.00.0.00.0.0.00.0.00.0.0.00.0.00.0.0.00.0.00.0.0.00.0.00.0.0.00

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

Cf. A060106, A060107, A081031, A081032.

Flatten[Table[{0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1}, {8}]][[1 ;; 88]] (* Jean-François Alcover, Apr 24 2017 *)
PadRight[{},88,{0,1,0,0,1,0,1,0,0,1,0,1}] (* Harvey P. Dale, Sep 14 2020 *)

a(n) = floor((5n+7)/12) - floor((5n+2)/12). - David W. Wilson, Mar 02 2003
G.f.: -x*(1+x^3+x^5+x^8+x^10) / ( (x-1) *(1+x+x^2) *(1+x) *(1-x+x^2) *(1+x^2) *(x^4-x^2+1) ). - R. J. Mathar, Dec 16 2016
a(n) = 0 if n is in A060107, a(n) = 1 if n is in A060106. - 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)

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

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.

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A059620 Colors of the 88 keys of the standard piano: white keys = 0, black keys = 1, 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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A081031 Positions of white 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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