|
%I #44 Sep 08 2022 08:45:10
%S 0,4,7,10,12,16,19,22,24,28,31,34,36,40,43,46,48,52,55,58,60,64,67,70,
%T 72,76,79,82,84,88,91,94,96,100,103,106,108,112,115,118,120,124,127,
%U 130,132,136,139,142,144,148,151,154,156,160,163,166,168,172
%N Numbers that are congruent to {0, 4, 7, 10} mod 12.
%C Key-numbers of the pitches of a dominant seventh chord on a standard chromatic keyboard, with root = 0.
%H G. C. Greubel, <a href="/A083032/b083032.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (1,0,0,1,-1).
%F G.f.: x^2*(4 + 3*x + 3*x^2 + 2*x^3)/((1 + x)*(1 + x^2)*(1 - x)^2). - _R. J. Mathar_, Oct 08 2011
%F From _Wesley Ivan Hurt_, May 19 2016: (Start)
%F a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
%F a(n) = (12*n - 9 + (-1)^n + (-1)^((n+1)/2) + (-1)^(-(n+1)/2))/4. (End)
%F a(2k) = A016957(k-1) for k > 0, a(2k-1) = A272975(k). - _Wesley Ivan Hurt_, Jun 01 2016
%F E.g.f.: (4 - sin(x) + (6*x - 5)*sinh(x) + (6*x - 4)*cosh(x))/2. - _Ilya Gutkovskiy_, Jun 01 2016
%F From _Jianing Song_, Sep 22 2018: (Start)
%F a(n) = (12*n - 9 + (-1)^n - 2*sin(n*Pi/2))/4.
%F a(n) = a(n-4) + 12 for n > 4. (End)
%F Sum_{n>=2} (-1)^n/a(n) = log(3)/8 - log(2)/12 + sqrt(3)*log(sqrt(3)+2)/12 - (5*sqrt(3)-6)*Pi/72. - _Amiram Eldar_, Dec 31 2021
%p A083032:=n->(12*n-9+(-1)^n+(-1)^((n+1)/2)+(-1)^(-(n+1)/2))/4: seq(A083032(n), n=1..100); # _Wesley Ivan Hurt_, May 19 2016
%t Select[Range[0,200], MemberQ[{0,4,7,10}, Mod[#,12]]&] (* _Harvey P. Dale_, Sep 13 2011 *)
%t LinearRecurrence[{1,0,0,1,-1},{0,4,7,10,12},100] (* _G. C. Greubel_, Jun 01 2016 *)
%o (Magma) [(12*n-9+(-1)^n+(-1)^((n+1) div 2)+(-1)^(-(n+1) div 2))/4: n in [1..100]]; // _Wesley Ivan Hurt_, May 19 2016
%o (PARI) my(x='x+O('x^99)); concat(0, Vec(x^2*(4+3*x+3*x^2+2*x^3)/((1+x)*(1+x^2)*(1-x)^2))) \\ _Altug Alkan_, Sep 21 2018
%o (GAP) Filtered([0..200],n-> n mod 12=0 or n mod 12=4 or n mod 12=7 or n mod 12=10); # _Muniru A Asiru_, Sep 22 2018
%Y Bisections: A016957, A272975.
%Y A guide for some sequences related to modes and chords:
%Y Modes:
%Y Lydian mode (F): A083089
%Y Ionian mode (C): A083026
%Y Mixolydian mode (G): A083120
%Y Dorian mode (D): A083033
%Y Aeolian mode (A): A060107 (raised seventh: A083028)
%Y Phrygian mode (E): A083034
%Y Locrian mode (B): A082977
%Y Chords:
%Y Major chord: A083030
%Y Minor chord: A083031
%Y Dominant seventh chord: this sequence
%K nonn,easy
%O 1,2
%A James Ingram (j.ingram(AT)t-online.de), Jun 01 2003
|
