A190974 - OEIS

%I #27 Jun 11 2022 03:33:53

%S 0,1,7,44,273,1691,10472,64849,401583,2486836,15399937,95365379,

%T 590557968,3657078881,22646762327,140241941884,868459781553,

%U 5378008761451,33303762422392,206236293149489,1277135239934463,7908765213793796,48975680296884257

%N a(n) = 7*a(n-1) - 5*a(n-2), with a(0)=0, a(1)=1.

%H G. C. Greubel, <a href="/A190974/b190974.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 (7,-5).

%F a(n) = ((7/2 + 1/2*sqrt(29))^n - (7/2 - 1/2*sqrt(29))^n)/sqrt(29). - _Giorgio Balzarotti_, May 28 2011

%F G.f.: x/(1 - 7*x + 5*x^2). - _Philippe Deléham_, Oct 12 2011

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

%F a(n) = 5^((n-1)/2)*ChebyshevU(n-1, 7/(2*sqrt(5))).

%F E.g.f.: (2/sqrt(29))*exp(7*x/2)*sinh(sqrt(29)*x/2). (End)

%t LinearRecurrence[{7,-5}, {0,1}, 50]

%o (Magma) [n le 2 select n-1 else 7*Self(n-1) - 5*Self(n-2): n in [1..51]]; // _G. C. Greubel_, Jun 11 2022

%o (SageMath) [lucas_number1(n,7,5) 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