A016051 - OEIS

%I #38 Nov 23 2024 05:46:51

%S 3,6,12,15,21,24,30,33,39,42,48,51,57,60,66,69,75,78,84,87,93,96,102,

%T 105,111,114,120,123,129,132,138,141,147,150,156,159,165,168,174,177,

%U 183,186,192,195,201,204,210,213,219,222,228,231,237,240

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

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (1,1,-1).

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

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

%F From _R. J. Mathar_, Dec 16 2009: (Start)

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

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

%F a(n) = 9*(n-1) - a(n-1) (with a(1)=3). - _Vincenzo Librandi_, Nov 19 2010

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

%F From _Amiram Eldar_, Nov 22 2024: (Start)

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

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

%t Select[Range[240], MatchQ[Mod[#, 9], 3|6]&] (* _Jean-François Alcover_, Sep 17 2013 *)

%t LinearRecurrence[{1,1,-1},{3,6,12},60] (* or *) #+{3,6}&/@(9*Range[0,30])//Flatten (* _Harvey P. Dale_, Oct 04 2021 *)

%Y Cf. A001651, A199589.

%Y Subsequence of A145204. - _Reinhard Zumkeller_, Oct 04 2008

%K nonn,easy

%O 1,1

%A _Robert G. Wilson v_