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%I #72 Jan 11 2024 08:33:35
%S 0,1,2,-1,-12,-19,22,139,168,-359,-1558,-1321,5148,16901,8062,-68381,
%T -177072,-12239,860882,1782959,-738492,-10391779,-17091098,17776699,
%U 121008888,153134281,-298775878,-1363223161,-1232566932
%N a(n) = A006496(n)/2.
%C Partial sums of A006495. - _Paul Barry_, Mar 16 2006
%C This is the Lucas U(P=2,Q=5) sequence. - _R. J. Mathar_, Oct 24 2012
%C With different signs, 0, 1, -2, -1, 12, -19, -22, 139, -168, -359, 1558, ... we obtain the Lucas U(-2,5) sequence. - _R. J. Mathar_, Jan 08 2013
%H Vincenzo Librandi, <a href="/A045873/b045873.txt">Table of n, a(n) for n = 0..500</a>
%H Ronald Orozco López, <a href="https://arxiv.org/abs/2211.04450">Deformed Differential Calculus on Generalized Fibonacci Polynomials</a>, arXiv:2211.04450 [math.CO], 2022.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Lucas_sequence">Lucas sequence</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (2,-5).
%H <a href="/index/Lu#Lucas">Index entries for Lucas sequences</a>
%F a(n)^2 = A094423(n).
%F From _Paul Barry_, Sep 20 2003: (Start)
%F O.g.f.: x/(1 - 2*x + 5*x^2).
%F E.g.f.: exp(x)*sin(2*x)/2.
%F a(n) = 2*a(n-1) - 5*a(n-2), a(0)=0, a(1)=1.
%F a(n) = ((1 + 2*i)^n - (1 - 2*i)^n)/(4*i), where i=sqrt(-1).
%F a(n) = Im{(1 + 2*i)^n/2}.
%F a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k+1)*(-4)^k. (End)
%F a(n+1) = Sum_{k=0..n} binomial(k,n-k)*2^k*(-5/2)^(n-k). - _Paul Barry_, Mar 16 2006
%F G.f.: 1/(4*x - 1/G(0)) where G(k) = 1 - (k+1)/(1 - x/(x - (k+1)/G(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Dec 06 2012
%F G.f.: Q(0)*x/2, where Q(k) = 1 + 1/(1 - x*(4*k+2 - 5*x)/( x*(4*k+4 - 5*x) + 1/Q(k+1) )); (continued fraction). - _Sergei N. Gladkovskii_, Aug 30 2013
%F a(n) = 5^((n-1)/2)*ChebyshevU(n-1, 1/sqrt(5)). - _G. C. Greubel_, Jan 11 2024
%p seq(coeff(series(x/(1-2*x+5*x^2),x,n+1), x, n), n = 0 .. 30); # _Muniru A Asiru_, Oct 23 2018
%t LinearRecurrence[{2,-5}, {0,1}, 40] (* _G. C. Greubel_, Jan 11 2024 *)
%o (GAP) a:=[0,1];; for n in [3..30] do a[n]:=2*a[n-1]-5*a[n-2]; od; a; # _Muniru A Asiru_, Oct 23 2018
%o (Magma) I:=[0,1]; [n le 2 select I[n] else 2*Self(n-1) - 5*Self(n-2): n in [1..50]]; // _G. C. Greubel_, Oct 22 2018
%o (PARI) concat(0,Vec(1/(1-2*x+5*x^2)+O(x^99))) \\ _Charles R Greathouse IV_, Dec 22 2011
%o (Sage) [lucas_number1(n,2,5) for n in range(0, 29)] # _Zerinvary Lajos_, Apr 23 2009
%o (SageMath)
%o A045873=BinaryRecurrenceSequence(2,-5,0,1)
%o [A045873(n) for n in range(41)] # _G. C. Greubel_, Jan 11 2024
%Y Cf. A006495, A006496, A084102, A088136, A088137, A088139, A094423.
%K sign,easy
%O 0,3
%A _N. J. A. Sloane_
%E More terms from _Paul Barry_, Sep 20 2003
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