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

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

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

Links

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

Crossrefs

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.

Programs

Magma

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

Mathematica

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

PARI

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

Formula

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