A190976 - OEIS

%I #26 Jun 11 2022 03:34:01

%S 0,1,8,61,464,3529,26840,204133,1552544,11807953,89805992,683024077,

%T 5194774640,39509124889,300488675192,2285382026869,17381590189376,

%U 132196575434401,1005427832907080,7646832936953437,58158379996906256,442326541164389737

%N a(n) = 8*a(n-1) - 3*a(n-2), with a(0)=0, a(1)=1.

%H G. C. Greubel, <a href="/A190976/b190976.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 (8,-3).

%F a(n) = ((4 + sqrt(13))^n - (4 - sqrt(13))^n)/(2*sqrt(13)). - _Giorgio Balzarotti_, May 28 2011

%F G.f.: x/(1 - 8*x + 3*x^2). - _Philippe Deléham_, Oct 12 2011

%F From _G. C. Greubel_, Jun 11 2022: (Start)

%F a(n) = 3^((n-1)/2)*ChebyshevU(n-1, 4/sqrt(3)).

%F E.g.f.: (1/sqrt(13))*exp(4*x)*sinh(sqrt(13)*x). (End)

%t LinearRecurrence[{8,-3}, {0,1}, 50]

%o (Magma) [n le 2 select n-1 else 8*Self(n-1) - 3*Self(n-2): n in [1..51]]; // _G. C. Greubel_, Jun 11 2022

%o (SageMath) [lucas_number1(n,8,3) for n in (0..50)] # _G. C. Greubel_, Jun 11 2022

%Y Cf. A190958 (index to generalized Fibonacci sequences).

%K nonn,easy

%O 0,3

%A _Vladimir Joseph Stephan Orlovsky_, May 24 2011