A082977 Numbers that are congruent to {0, 1, 3, 5, 6, 8, 10} mod 12.

0, 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, 66, 68, 70, 72, 73, 75, 77, 78, 80, 82, 84, 85, 87, 89, 90, 92, 94, 96, 97, 99, 101, 102, 104, 106, 108, 109, 111

Offset

1,3

Comments

James Ingram suggests that this (with the initial 0 omitted) is the correct version of Fludd's sequence, rather than A047329. See also A083026.

Key-numbers of the pitches of a Hypophrygian mode scale on a standard chromatic keyboard, with root = 0. A Hypophrygian mode scale can, for example, be played on consecutive white keys of a standard keyboard, starting on the root tone B. - James Ingram (j.ingram(AT)t-online.de), Jun 01 2003

References

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

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

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 (1,0,0,0,0,0,1,-1).

Formula

G.f.: x*(1 + x + 2*x^4)*(1 + x + x^2)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)). - R. J. Mathar, Sep 17 2008

From Wesley Ivan Hurt, Jul 19 2016: (Start)

a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.

a(n) = (84*n - 105 - 2*(n mod 7) - 2*((n + 1) mod 7) + 5*((n + 2) mod 7) - 2*((n + 3) mod 7) - 2*((n + 4) mod 7) + 5*((n + 5) mod 7) - 2*((n + 6) mod 7))/49.

a(7k) = 12k - 2, a(7k-1) = 12k - 4, a(7k-2) = 12k - 6, a(7k-3) = 12k - 7, a(7k-4) = 12k - 9, a(7k-5) = 12k - 11, a(7k-6) = 12k - 12. (End)

a(n) = a(n-7) + 12 for n > 7. - Jianing Song, Sep 22 2018

a(n) = floor(12*(n-1)/7). - Federico Provvedi, Oct 18 2018

Maple

A082977:=n->12*floor(n/7)+[0, 1, 3, 5, 6, 8, 10][(n mod 7)+1]: seq(A082977(n), n=0..100); # Wesley Ivan Hurt, Jul 19 2016

Mathematica

CoefficientList[Series[x(1 + x + 2*x^4)(1 + x + x^2)/((1 - x)^2*(1 + x + x^2 + x^3 + x^4 + x^5 + x^6)), {x, 0, 100}], x] (* Vincenzo Librandi, Jan 06 2013 *)
fQ[n_] := MemberQ[{0, 1, 3, 5, 6, 8, 10}, Mod[n, 12]]; Select[ Range[0, 111], fQ] (* Robert G. Wilson v, Jan 07 2014 *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 1, 3, 5, 6, 8, 10, 12}, 70] (* Jianing Song, Sep 22 2018 *)
Floor@Range[0, 2^8, 12/7] (* Federico Provvedi, Oct 18 2018 *)

Prog

(Haskell)
a082977 n = a082977_list !! (n-1)
a082977_list = [0, 1, 3, 5, 6, 8, 10] ++ map (+ 12) a082977_list
-- Reinhard Zumkeller, Jan 07 2014
(Magma) [n : n in [0..150] | n mod 12 in [0, 1, 3, 5, 6, 8, 10]]; // Wesley Ivan Hurt, Jul 19 2016
(PARI) x='x+O('x^99); concat(0, Vec(x*(1+x+2*x^4)*(1+x+x^2)/((1-x)^2*(1+x+x^2+x^3+x^4+x^5+x^6)))) \\ Jianing Song, Sep 22 2018

Crossrefs

Cf. A047329. Different from A000210.

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): this sequence

Chords:

Major chord: A083030

Minor chord: A083031

Dominant seventh chord: A083032

Sequence in context: A299233 A320997 A083042 * A000210 A329829 A182760

Adjacent sequences: A082974 A082975 A082976 * A082978 A082979 A082980

Keyword

nonn,easy

Author

N. J. A. Sloane, May 31 2003

Status

approved