A190971 - OEIS

%I #32 Jun 11 2022 06:32:36

%S 0,1,5,15,25,-25,-375,-1625,-4375,-5625,15625,134375,515625,1234375,

%T 1015625,-7265625,-46484375,-159765625,-333984375,-72265625,

%U 2978515625,15615234375,48291015625,85302734375,-56396484375,-1135009765625,-5111083984375,-14205322265625

%N a(n) = 5*a(n-1) - 10*a(n-2), with a(0)=0, a(1)=1.

%H G. C. Greubel, <a href="/A190971/b190971.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 (5,-10).

%F a(n) = (i/sqrt(15))*(((5 - i*sqrt(15))/2)^n - ((5 + i*sqrt(15))/2)^n). - _Giorgio Balzarotti_, May 28 2011

%F G.f.: x/(1 - 5*x + 10*x^2). - _Philippe Deléham_, Oct 12 2011

%F From _G. C. Greubel_, Jun 10 2022: (Start)

%F a(n) = 10^((n-1)/2) * ChebyshevU(n-1, sqrt(10)/4).

%F E.g.f.: (2/sqrt(15))*exp(5*x/2)*sin(sqrt(15)*x/2). (End)

%t LinearRecurrence[{5,-10}, {0,1}, 50]

%o (PARI) concat(0, Vec(x/(1-5*x+10*x^2) + O(x^100))) \\ _Altug Alkan_, Nov 26 2015

%o (Magma) [n le 2 select n-1 else 5*(Self(n-1) - 2*Self(n-2)): n in [1..51]]; // _G. C. Greubel_, Jun 10 2022

%o (Sage) [lucas_number1(n,5,10) for n in (0..50)] # _G. C. Greubel_, Jun 10 2022

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

%K sign

%O 0,3

%A _Vladimir Joseph Stephan Orlovsky_, May 24 2011