A083881 - OEIS

%I #22 Sep 08 2022 08:45:10

%S 1,3,12,54,252,1188,5616,26568,125712,594864,2814912,13320288,

%T 63032256,298271808,1411437312,6678993024,31605334272,149558047488,

%U 707716279296,3348949390848,15847398669312,74990695670784,354859782008832

%N a(n) = 6*a(n-1) - 6*a(n-2), with a(0)=1, a(1)=3.

%C Binomial transform of A001075.

%H G. C. Greubel, <a href="/A083881/b083881.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 (6,-6).

%F a(n) = ((3-sqrt(3))^n + (3+sqrt(3))^n)/2.

%F a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2*k)*3^(n-k).

%F G.f.: (1-3*x)/(1-6*x+6*x^2).

%F E.g.f.: exp(3*x) * cosh(x*sqrt(3)).

%F a(n) = right and left terms in M^n * [1 1 1] where M = the 3X3 matrix [1 1 1 / 1 4 1 / 1 1 1]. M^n * [1 1 1] = [a(n) A030192(n) a(n)]. E.g. a(3) = 54 since M^3 * [1 1 1] = [54 144 54] = [a(3) A030192(3) a(3)]. - _Gary W. Adamson_, Dec 18 2004

%F a(n) = Sum_{k, 0<=k<=n}3^k*A098158(n,k). - _Philippe Deléham_, Dec 04 2006

%F G.f.: A(x) = G(0) where G(k) =  1 + 3*x/((1-3*x) - x*(1-3*x)/(x + (1-3*x)/G(k+1) )); (recursively defined continued fraction). - _Sergei N. Gladkovskii_, Dec 29 2012.

%t f[n_]:= Simplify[(3 + Sqrt@3)^n + (3 - Sqrt@3)^n]/2; Array[f, 30, 0] (* _Robert G. Wilson v_, Oct 31 2010 *)

%t LinearRecurrence[{6,-6}, {1,3}, 30] (* _G. C. Greubel_, Aug 01 2019 *)

%o (PARI) my(x='x+O('x^30)); Vec((1-3*x)/(1-6*x+6*x^2)) \\ _G. C. Greubel_, Aug 01 2019

%o (Magma) I:=[1,3]; [n le 2 select I[n] else 6*Self(n-1) -6*Self(n-2): n in [1..30]]; // _G. C. Greubel_, Aug 01 2019

%o (Sage) ((1-3*x)/(1-6*x+6*x^2)).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Aug 01 2019

%o (GAP) a:=[1,3];; for n in [3..30] do a[n]:=6*a[n-1]-6*a[n-2]; od; a; # _G. C. Greubel_, Aug 01 2019

%Y Cf. A083882.

%Y Cf. A030192.

%K easy,nonn

%O 0,2

%A _Paul Barry_, May 08 2003