A291454 -id:A291454 - cp's OEIS frontend

A083026 Numbers that are congruent to {0, 2, 4, 5, 7, 9, 11} mod 12.

0, 2, 4, 5, 7, 9, 11, 12, 14, 16, 17, 19, 21, 23, 24, 26, 28, 29, 31, 33, 35, 36, 38, 40, 41, 43, 45, 47, 48, 50, 52, 53, 55, 57, 59, 60, 62, 64, 65, 67, 69, 71, 72, 74, 76, 77, 79, 81, 83, 84, 86, 88, 89, 91, 93, 95, 96, 98, 100, 101, 103, 105, 107, 108, 110

Offset: 1

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

Key-numbers of the pitches of a major scale on a standard chromatic keyboard, with root = 0.
Also key-numbers of the pitches of an Ionian mode scale on a standard chromatic keyboard, with root = 0. An Ionian mode scale can, for example, be played on consecutive white keys of a standard keyboard, starting on the root tone C.
Cumulative sum of A291454. - Halfdan Skjerning, Aug 30 2017

G. C. Greubel, Table of n, a(n) for n = 1..1000
Wikipedia, Scale Music
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A291454.
A guide for some sequences related to modes and chords:
Modes:
Lydian mode (F): A083089
Ionian mode (C): this sequence
Mixolydian mode (G): A083120
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, 11]]; // Wesley Ivan Hurt, Jul 20 2016

A083026:=n->12*floor(n/7)+[0, 2, 4, 5, 7, 9, 11][(n mod 7)+1]: seq(A083026(n), n=0..100); # Wesley Ivan Hurt, Jul 20 2016

Select[Range[0, 150], MemberQ[{0, 2, 4, 5, 7, 9, 11}, Mod[#, 12]] &] (* Wesley Ivan Hurt, Jul 20 2016 *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 2, 4, 5, 7, 9, 11, 12}, 70] (* Jianing Song, Sep 22 2018 *)
Quotient[12*Range[60], 7] - 1 (* Federico Provvedi, Sep 10 2022 *)

a(n)=[-1, 0, 2, 4, 5, 7, 9][n%7+1] + n\7*12 \\ Charles R Greathouse IV, Jul 20 2016

x='x+O('x^99); concat(0, Vec(x^2*(x+1)*(x^5+x^4+x^3+x^2+2)/((x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2))) \\ Jianing Song, Sep 22 2018

G.f.: x^2*(x + 1)*(x^5 + x^4 + x^3 + x^2 + 2)/((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 - 70 - 2*(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) + 5*((n + 6) mod 7))/49.
a(7k) = 12k - 1, 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

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

cp's OEIS Frontend

A083026 Numbers that are congruent to {0, 2, 4, 5, 7, 9, 11} mod 12.

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