A084120 - OEIS

%I #31 Oct 14 2022 08:55:18

%S 1,3,15,81,441,2403,13095,71361,388881,2119203,11548575,62933841,

%T 342957321,1868942403,10184782455,55501867521,302456857761,

%U 1648235544003,8982042690735,48947549512401,266739169002201

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

%C Binomial transform of A084059.

%H Harvey P. Dale, <a href="/A084120/b084120.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,-3).

%F a(n) = ((3+sqrt(6))^n + (3-sqrt(6))^n)/2.

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

%F E.g.f.: exp(3*x)*cosh(sqrt(6)*x).

%F a(n) = 3^n * Sum_{k=0..floor(n/2)} C(n, 2*k)*(2/3)^k. - _Paul Barry_, Sep 10 2005

%F Lim_{n -> oo} a(n)/a(n-1) = (3 + sqrt(6)) = 5.445489742... - _Gary W. Adamson_, Mar 19 2008

%F a(n) = Sum_{k=0..n} A147720(n,k)*3^k. - _Philippe Deléham_, Nov 15 2008

%F G.f.: G(0)/2, where G(k)= 1 + 1/(1 - x*(2*k-3)/(x*(2*k-1) - 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 27 2013

%F a(n) = A138395(n) - 3*A138395(n-1). - _R. J. Mathar_, May 11 2022

%e G.f. = 1 + 3*x + 15*x^2 + 81*x^3 + 441*x^4 + 2403*x^5 + 13095*x^6 + ...

%t LinearRecurrence[{6,-3},{1,3},30] (* _Harvey P. Dale_, Feb 25 2014 *)

%o (PARI) {a(n) = if( n<0, 0, polsym(x^2 - 6*x + 3, n)[1+n] / 2)};

%o (Sage) [lucas_number2(n,6,3)/2 for n in range(0,27)] # _Zerinvary Lajos_, Jul 08 2008

%o (Magma) [n le 2 select 3^(n-1) else 6*Self(n-1) -3*Self(n-2): n in [1..41]]; // _G. C. Greubel_, Oct 13 2022

%Y Cf. A084059, A138395, A147720.

%K easy,nonn

%O 0,2

%A _Paul Barry_, May 13 2003