A026565 - OEIS

%I #24 Sep 08 2022 08:44:49

%S 1,3,9,18,54,108,324,648,1944,3888,11664,23328,69984,139968,419904,

%T 839808,2519424,5038848,15116544,30233088,90699264,181398528,

%U 544195584,1088391168,3265173504,6530347008,19591041024,39182082048

%N a(n) = 6*a(n-2), starting with 1, 3, 9.

%H G. C. Greubel, <a href="/A026565/b026565.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (0,6).

%F a(n) = Sum_{j=0..2*n} A026552(n, j).

%F G.f.: (1+3*x+3*x^2)/(1-6*x^2). - _Ralf Stephan_, Feb 03 2004

%F a(0)=1, a(1)=3; a(n) = 3*a(n-1) if n is even, a(n) = 2*a(n-1) if n is odd. - _Vincenzo Librandi_, Nov 19 2010

%F a(n) = (1/4)*6^(n/2)*(3*(1+(-1)^n) + sqrt(6)*(1-(-1)^n)) - (1/2)*[n=0]. - _G. C. Greubel_, Dec 17 2021

%t Table[(1/4)*6^(n/2)*(3*(1+(-1)^n) + Sqrt[6]*(1-(-1)^n)) - (1/2)*Boole[n==0], {n, 0, 35}] (* _G. C. Greubel_, Dec 17 2021 *)

%o (Magma) [1] cat [n le 2 select 3^n else 6*Self(n-2): n in [1..35]]; // _G. C. Greubel_, Dec 17 2021

%o (Sage)

%o def A026565(n): return ( (3/2)*6^(n/2) if (n%2==0) else 3*6^((n-1)/2) ) - bool(n==0)/2

%o [A026565(n) for n in (0..30)] # _G. C. Greubel_, Dec 17 2021

%Y Cf. A026532, A026534, A026551, A026552, A026567.

%K nonn,easy

%O 0,2

%A _Clark Kimberling_

%E Better name from _Ralf Stephan_, Jul 17 2013