A110569 -id:A110569 - cp's OEIS frontend

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

1, 4, 5, 7, 9, 10, 13, 16, 17, 19, 21, 22, 25, 28, 29, 31, 33, 34, 37, 40, 41, 43, 45, 46, 49, 52, 53, 55, 57, 58, 61, 64, 65, 67, 69, 70, 73, 76, 77, 79, 81, 82, 85, 88, 89, 91, 93, 94, 97, 100, 101, 103, 105, 106, 109, 112, 113, 115, 117, 118, 121, 124, 125

Offset: 1

Bob Selcoe, Jan 09 2016

Definitions and explanation:
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.
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), kFirst 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.
For kA016777) 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.
These are the numbers congruent to {1, 4, 5, 7, 9, 10} mod 12. - N. J. A. Sloane, Feb 06 2016

			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

Robert Walker, Bounce Metronome. See example 4:3 for the audio-visual representation of this sequence.
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-1).

Cf. A016777, A016813, A110569.

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

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

Select[Range@ 125, Or[Mod[#, 3] == 1, Mod[#, 4] == 1] &] (* or *)
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 *)
LinearRecurrence[{2, -2, 2, -2, 2, -1}, {1, 4, 5, 7, 9, 10}, 70] (* Vincenzo Librandi, Jan 10 2016 *)

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

From Colin Barker, Jan 09 2016: (Start)
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.
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)
From Wesley Ivan Hurt, Jun 29 2016: (Start)
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.
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)

A110568 Period 6: repeat [1, 0, 2, 2, 0, 1].

Original entry on oeis.org

1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2, 2, 0, 1, 1, 0, 2

Offset: 0

Paul Barry, Jul 27 2005

Permutation of {0, 1, 2}, followed by its reversal, repeated.

Index entries for linear recurrences with constant coefficients, signature (1,-1,1,-1,1).

Cf. A078008, A088689, A105198, A110549, A110550, A110551, A110569.

&cat [[1, 0, 2, 2, 0, 1]^^30]; // Wesley Ivan Hurt, Jun 28 2016

A110568:=n->[1, 0, 2, 2, 0, 1][(n mod 6)+1]: seq(A110568(n), n=0..100); # Wesley Ivan Hurt, Jun 28 2016

Mod[#,3]&/@CoefficientList[Series[(1-x)/(1-x-2x^2),{x,0,100}],x] (* Harvey P. Dale, Mar 30 2011 *)
PadRight[{}, 100, {1, 0, 2, 2, 0, 1}] (* Wesley Ivan Hurt, Jun 28 2016 *)
LinearRecurrence[{1,-1,1,-1,1},{1,0,2,2,0},100] (* Harvey P. Dale, Apr 03 2019 *)

x='x+O('x^50); Vec((1-x+3*x^2-x^3+x^4)/(1-x+x^2-x^3+x^4-x^5)) \\ G. C. Greubel, Aug 31 2017

a(n) = A078008(n) mod 3.
G.f.: (1-x+3*x^2-x^3+x^4) / (1-x+x^2-x^3+x^4-x^5).
a(n) = a(n-1) - a(n-2) + a(n-3) - a(n-4) + a(n-5) for n>4.
a(n) = 1 + cos(2*Pi*n/3)/2 - sqrt(3)*sin(2*Pi*n/3)/2 - cos(Pi*n/3)/2 + sqrt(3)*sin(Pi*n/3)/6.
a(n) = a(n-6) for n > 5. - Wesley Ivan Hurt, Jun 28 2016
a(n) = ((n-1)*(-1)^(n-1) mod 3). - Wesley Ivan Hurt, Jan 07 2021

Name changed by Wesley Ivan Hurt, Jun 28 2016

Showing 1-2 of 2 results.

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula