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A319280
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Numbers that are congruent to {0, 4, 7, 11} mod 12.
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4
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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
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OFFSET
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1,2
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COMMENTS
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Key-numbers of the pitches of a major seventh chord on a standard chromatic keyboard, with root = 0.
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LINKS
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FORMULA
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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
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MATHEMATICA
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Select[Range[0, 200], MemberQ[{0, 4, 7, 11}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 4, 7, 11, 12}, 100]
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PROG
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(Magma) [n : n in [0..150] | n mod 12 in [0, 4, 7, 11]]
(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)))
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CROSSREFS
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A guide for some sequences related to modes and chords:
Modes:
Third chords:
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
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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