A016051 Numbers of the form 9*k+3 or 9*k+6.

3, 6, 12, 15, 21, 24, 30, 33, 39, 42, 48, 51, 57, 60, 66, 69, 75, 78, 84, 87, 93, 96, 102, 105, 111, 114, 120, 123, 129, 132, 138, 141, 147, 150, 156, 159, 165, 168, 174, 177, 183, 186, 192, 195, 201, 204, 210, 213, 219, 222, 228, 231, 237, 240

Offset: 1

Author

Robert G. Wilson v

Keywords

• nonn
• easy

Links

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

Crossrefs

Cf. A001651, A199589.

Subsequence of A145204. - Reinhard Zumkeller, Oct 04 2008

Programs

Mathematica

Select[Range[240], MatchQ[Mod[#, 9], 3|6]&] (* Jean-François Alcover, Sep 17 2013 *)
LinearRecurrence[{1,1,-1},{3,6,12},60] (* or *) #+{3,6}&/@(9*Range[0,30])//Flatten (* Harvey P. Dale, Oct 04 2021 *)

Formula

a(n) = 3*A001651(n).

a(n+1) = a(n) + its digital root in decimal base.

From R. J. Mathar, Dec 16 2009: (Start)

a(n) = a(n-1) + a(n-2) - a(n-3) = 9*n/2 - 9/4 - 3*(-1)^n/4.

G.f: 3*x*(1+x+x^2)/((1+x)*(x-1)^2). (End)

a(n) = 9*(n-1) - a(n-1) (with a(1)=3). - Vincenzo Librandi, Nov 19 2010

Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/(9*sqrt(3)). - Amiram Eldar, Sep 26 2022

From Amiram Eldar, Nov 22 2024: (Start)

Product_{n>=1} (1 - (-1)^n/a(n)) = (2/sqrt(3)) * cos(Pi/18) (A199589).

Product_{n>=1} (1 + (-1)^n/a(n)) = (2/sqrt(3)) * sin(2*Pi/9). (End)