A109007 a(n) = gcd(n,3).

3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1, 1, 3, 1

Offset: 0

Author

Mitch Harris

Keywords

• nonn
• easy
• mult

Comments

For n>1: a(n) = GCD of the n-th and (n+2)-th triangular numbers = A050873(A000217(n+2), A000217(n)). - Reinhard Zumkeller, May 28 2007

From Klaus Brockhaus, May 24 2010: (Start)

Continued fraction expansion of (3+sqrt(17))/2.

Decimal expansion of 311/999. (End)

Links

• Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Crossrefs

Cf. A000217, A026741, A050873, A109004, A130334.

Cf. A178255 (decimal expansion of (3+sqrt(17))/2). - Klaus Brockhaus, May 24 2010

Programs

Magma

[Gcd(n,3) : n in [0..100]]; // Wesley Ivan Hurt, Jul 24 2016

Maple

A109007:=n->gcd(n,3): seq(A109007(n), n=0..100); # Wesley Ivan Hurt, Jul 24 2016

Mathematica

GCD[Range[0,100],3] (* or *) PadRight[{},110,{3,1,1}] (* Harvey P. Dale, Jun 28 2015 *)

PARI

a(n)=gcd(n,3) \\ Charles R Greathouse IV, Sep 24 2015

Formula

a(n) = 1 + 2*[3|n] = 1 + 2(1 + 2*cos(2*n*Pi/3))/3, where [x|y] = 1 when x divides y, 0 otherwise.

a(n) = a(n-3) for n>2.

Multiplicative with a(p^e, 3) = gcd(p^e, 3). - David W. Wilson, Jun 12 2005

O.g.f.: -(3+x+x^2)/((x-1)*(x^2+x+1)). - R. J. Mathar, Nov 24 2007

Dirichlet g.f. zeta(s)*(1+2/3^s). - R. J. Mathar, Apr 08 2011

a(n) = 2*floor(((n-1) mod 3)/2) + 1. - Gary Detlefs, Dec 28 2011

a(n) = 3^(1 - sgn(n mod 3)). - Wesley Ivan Hurt, Jul 24 2016

a(n) = 3/(1 + 2*((n^2) mod 3)). - Timothy Hopper, Feb 25 2017

a(n) = (5 + 4*cos(2*n*Pi/3))/3. - Wesley Ivan Hurt, Oct 04 2018