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A267027 "Polyrhythmic sequence" P(3,4): numbers congruent to 1 mod 3 (A016777) or 1 mod 4 (A016813).

%I A267027 #53 Sep 08 2022 08:46:15
%S A267027 1,4,5,7,9,10,13,16,17,19,21,22,25,28,29,31,33,34,37,40,41,43,45,46,
%T A267027 49,52,53,55,57,58,61,64,65,67,69,70,73,76,77,79,81,82,85,88,89,91,93,
%U A267027 94,97,100,101,103,105,106,109,112,113,115,117,118,121,124,125
%N A267027 "Polyrhythmic sequence" P(3,4): numbers congruent to 1 mod 3 (A016777) or 1 mod 4 (A016813).
%C A267027 Definitions and explanation:
%C A267027 Let {S} be a multi-element set of increasing integers > 1, where gcd(S)=1 and no element s(k), k=1..z, is a multiple of any other.
%C A267027 Define a "Polyrhythmic sequence" P(S) to be a subsequence B_k == 1 (mod lcm(S)/s(k)), b_k(1)=1, combined. It is so named because the terms in P(S) reflect where the "beats" in music are played in a "s(z) against s(k), k<z" polyrhythm; terms in a complementary sequence to P(S) reflect "rests".
%C A267027 First differences are periodic and palindromic, with periods beginning at a(n) == 1 (mod lcm(S)). In this example, P(3,4): s(1)=3, s(2)=s(z)=4; lcm(3,4)=12, 12/s(1) = 4 and 12/s(2) = 3. So B_1 == 1 mod 4 (A016813) and B_2 == 1 mod 3 (A016777); first differences are cycle [3,1,2,2,1,3] (A110569) and periods repeat at a(n) == 1 mod 12.
%C A267027 For k<z, polyrhythms in music are described as "s(z) against s(k)" because s(k) beats for each separate rhythm (reflected by terms in each individual B_k subsequence) occur in the same lcm(S)-beat period. So P(3,4) is analogous to a "4 against 3" polyrhythm: a 4-beat rhythm (beats at B_2 = A016777) occurring in the same 12-beat period as a 3-beat rhythm (beats at B_1 = A016813). See link to "Robert Walker's Bounce Metronome", example 4:3, for an audio-visual representation.
%C A267027 These are the numbers congruent to {1, 4, 5, 7, 9, 10} mod 12. - _N. J. A. Sloane_, Feb 06 2016
%H A267027 Robert Walker, <a href="http://bouncemetronome.com/video-resources/polyrhythms/4-3-type-polyrhythms">Bounce Metronome</a>. See example 4:3 for the audio-visual representation of this sequence.
%H A267027 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (2,-2,2,-2,2,-1).
%F A267027 From _Colin Barker_, Jan 09 2016: (Start)
%F A267027 a(n) = 2*a(n-1)-2*a(n-2)+2*a(n-3)-2*a(n-4)+2*a(n-5)-a(n-6) for n>6.
%F A267027 G.f.: x*(1+2*x-x^2+3*x^3-x^4+2*x^5) / ((1-x)^2*(1-x+x^2)*(1+x+x^2)). (End)
%F A267027 From _Wesley Ivan Hurt_, Jun 29 2016: (Start)
%F A267027 a(n) = (12*n - 6 - 3*cos(n*Pi/3) - 3*cos(2*n*Pi/3) + sqrt(3)*sin(n*Pi/3) - sqrt(3)*sin(2*n*Pi/3))/6.
%F A267027 a(6k) = 12k-2, a(6k-1) = 12k-3, a(6k-2) = 12k-5, a(6k-3) = 12k-7, a(6k-4) = 12k-8, a(6k-5) = 12k-11. (End)
%e A267027 Let the "beats" (b) be the sounds and the "rests" (r) be the silences as heard when a "4 against 3" polyrhythm is played. (See link to "Robert Walker's Bounce Metronome", example 4:3). So we have b-r-r-b-b-r-b-r-b-b-r-r repeated; when repeated indefinitely, the beats are on 1,4,5,7,9,10 / 13,16,17,19,21,22 / 25,28,29,31,33,34... which combines numbers congruent to 1 mod 3 and 1 mod 4. - _Bob Selcoe_, Feb 04 2016
%p A267027 A267027:=n->12*floor(n/6)+[1, 4, 5, 7, 9, 10][(n mod 6)+1]: seq(A267027(n), n=0..100); # _Wesley Ivan Hurt_, Jun 29 2016
%t A267027 Select[Range@ 125, Or[Mod[#, 3] == 1, Mod[#, 4] == 1] &] (* or *)
%t A267027 CoefficientList[Series[x (1 + 2 x - x^2 + 3 x^3 - x^4 + 2 x^5)/((1 - x)^2 (1 - x + x^2) (1 + x + x^2)), {x, 0, 63}], x] (* _Michael De Vlieger_, Jan 09 2016 *)
%t A267027 LinearRecurrence[{2, -2, 2, -2, 2, -1}, {1, 4, 5, 7, 9, 10}, 70] (* _Vincenzo Librandi_, Jan 10 2016 *)
%o A267027 (PARI) Vec(x*(1+2*x-x^2+3*x^3-x^4+2*x^5)/((1-x)^2*(1-x+x^2)*(1+x+x^2)) + O(x^100)) \\ _Colin Barker_, Jan 09 2016
%o A267027 (Magma) I:=[1,4,5,7,9,10,13]; [n le 7 select I[n] else 2*Self(n-1)-2*Self(n-2)+2*Self(n-3)-2*Self(n-4)+2*Self(n-5)-Self(n-6): n in [1..70]]; // _Vincenzo Librandi_, Jan 10 2016
%Y A267027 Cf. A016777, A016813, A110569.
%K A267027 nonn,easy
%O A267027 1,2
%A A267027 _Bob Selcoe_, Jan 09 2016