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

Author

Bob Selcoe, Jan 09 2016

Keywords

• nonn
• easy

Comments

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), k

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.

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

Examples

			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
		

Links

• 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).

Crossrefs

Cf. A016777, A016813, A110569.

Programs

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

Maple

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

Mathematica

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 *)

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

Formula

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)

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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