A267027 -id:A267027 - cp's OEIS frontend

A047238 Numbers that are congruent to {0, 2} mod 6.

0, 2, 6, 8, 12, 14, 18, 20, 24, 26, 30, 32, 36, 38, 42, 44, 48, 50, 54, 56, 60, 62, 66, 68, 72, 74, 78, 80, 84, 86, 90, 92, 96, 98, 102, 104, 108, 110, 114, 116, 120, 122, 126, 128, 132, 134, 138, 140, 144, 146, 150, 152, 156, 158, 162

Offset: 1

N. J. A. Sloane

Complement of A047251, or "Polyrhythmic Sequence" P(2,3); the present sequence represents where the "rests" occur in a "3 against 2" polyrhythm. (See A267027 for definition and description). - Bob Selcoe, Jan 12 2016

Bruno Berselli, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,1,-1).

Cf. A047270 [(6*n-(-1)^n-1)/2], A047235 [(6*n-(-1)^n-3)/2], A047241 [(6*n-(-1)^n-5)/2].

Cf. A047251, A267027.

[n: n in [0..200]|n mod 6 in {0,2}]; // Vincenzo Librandi, Jan 12 2016

Select[Range[0,200],MemberQ[{0,2},Mod[#,6]]&] (* or *) LinearRecurrence[ {1,1,-1},{0,2,6},70] (* Harvey P. Dale, Jun 15 2011 *)

forstep(n=0,200,[2,4],print1(n", ")) \\ Charles R Greathouse IV, Oct 17 2011

From Bruno Berselli, Jun 24 2010: (Start)
G.f.: 2*x*(1+2*x)/((1+x)*(1-x)^2).
a(n) = a(n-1) + a(n-2) - a(n-3), a(0)=0, a(1)=2, a(2)=6.
a(n) = (6*n - (-1)^n-7)/2.
a(n) = 2*A032766(n-1). (End)
a(n) = 6*n - a(n-1) - 10 (with a(1)=0). - Vincenzo Librandi, Aug 05 2010
a(n+1) = Sum_{k>=0} A030308(n,k)*A111286(k+2). - Philippe Deléham, Oct 17 2011
a(n) = 2*floor(3*n/2). - Enrique Pérez Herrero, Jul 04 2012
Sum_{n>=2} (-1)^n/a(n) = sqrt(3)*Pi/36 + log(3)/4. - Amiram Eldar, Dec 13 2021
E.g.f: 3*(x-1)*exp(x) - cosh(x) + 4. - David Lovler, Jul 11 2022

A047251 Numbers that are congruent to {1, 3, 4, 5} (mod 6).

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 10, 11, 13, 15, 16, 17, 19, 21, 22, 23, 25, 27, 28, 29, 31, 33, 34, 35, 37, 39, 40, 41, 43, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 63, 64, 65, 67, 69, 70, 71, 73, 75, 76, 77, 79, 81, 82, 83, 85, 87, 88, 89, 91, 93, 94, 95, 97, 99

Offset: 1

N. J. A. Sloane

"Polyrhythmic Sequence" P(2,3): numbers congruent to 1 (mod 2) and 1 (mod 3). (See A267027 for definition and description.) - Bob Selcoe, Jan 12 2016

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

Cf. A016777, A047270, A267027.

[n: n in [0..150]|n mod 6 in {1,3,4,5}]; // Vincenzo Librandi, Jan 12 2016

A047251:=n->(-2-(-I)^n-I^n+6*n)/4: seq(A047251(n), n=1..100); # Wesley Ivan Hurt, May 31 2016

Select[Range[0, 200], MemberQ[{1, 3, 4, 5}, Mod[#, 6]] &] (* Vincenzo Librandi, Jan 12 2016 *)
LinearRecurrence[{2,-2,2,-1},{1,3,4,5},70] (* Harvey P. Dale, Feb 27 2024 *)

a(n) = (-2-(-I)^n-I^n+6*n)/4 \\ Colin Barker, Oct 19 2015

Vec(x*(x^3+x+1)/((x-1)^2*(x^2+1)) + O(x^100)) \\ Colin Barker, Oct 19 2015

From R. J. Mathar, Oct 08 2011: (Start)
a(n) = 3*n/2 - 1/2 - cos(Pi*n/2)/2.
G.f.: x*(x^3+x+1)/((x-1)^2*(x^2+1)). (End)
a(n) = (-2 - (-i)^n - i^n + 6n)/4, with i=sqrt(-1). - Colin Barker, Oct 19 2015
From Wesley Ivan Hurt, May 31 2016: (Start)
a(n) = 2*a(n-1) - 2*a(n-2) + 2*a(n-3) - a(n-4) for n>4.
a(2k) = A047270(k), a(2k-1) = A016777(k-1) for n>0. (End)
Sum_{n>=1} (-1)^(n+1)/a(n) = 5*sqrt(3)*Pi/36 - log(2)/3 + log(3)/4. - Amiram Eldar, Dec 17 2021

cp's OEIS Frontend

A047238 Numbers that are congruent to {0, 2} mod 6.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A047251 Numbers that are congruent to {1, 3, 4, 5} (mod 6).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula