A045873 a(n) = A006496(n)/2.
0, 1, 2, -1, -12, -19, 22, 139, 168, -359, -1558, -1321, 5148, 16901, 8062, -68381, -177072, -12239, 860882, 1782959, -738492, -10391779, -17091098, 17776699, 121008888, 153134281, -298775878, -1363223161, -1232566932
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..500
- Ronald Orozco López, Deformed Differential Calculus on Generalized Fibonacci Polynomials, arXiv:2211.04450 [math.CO], 2022.
- Wikipedia, Lucas sequence
- Index entries for linear recurrences with constant coefficients, signature (2,-5).
- Index entries for Lucas sequences
Programs
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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
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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
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Maple
seq(coeff(series(x/(1-2*x+5*x^2),x,n+1), x, n), n = 0 .. 30); # Muniru A Asiru, Oct 23 2018
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Mathematica
LinearRecurrence[{2,-5}, {0,1}, 40] (* G. C. Greubel, Jan 11 2024 *) -
PARI
concat(0,Vec(1/(1-2*x+5*x^2)+O(x^99))) \\ Charles R Greathouse IV, Dec 22 2011
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Sage
[lucas_number1(n,2,5) for n in range(0, 29)] # Zerinvary Lajos, Apr 23 2009
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SageMath
A045873=BinaryRecurrenceSequence(2,-5,0,1) [A045873(n) for n in range(41)] # G. C. Greubel, Jan 11 2024
Formula
a(n)^2 = A094423(n).
From Paul Barry, Sep 20 2003: (Start)
O.g.f.: x/(1 - 2*x + 5*x^2).
E.g.f.: exp(x)*sin(2*x)/2.
a(n) = 2*a(n-1) - 5*a(n-2), a(0)=0, a(1)=1.
a(n) = ((1 + 2*i)^n - (1 - 2*i)^n)/(4*i), where i=sqrt(-1).
a(n) = Im{(1 + 2*i)^n/2}.
a(n) = Sum_{k=0..floor(n/2)} binomial(n, 2k+1)*(-4)^k. (End)
a(n+1) = Sum_{k=0..n} binomial(k,n-k)*2^k*(-5/2)^(n-k). - Paul Barry, Mar 16 2006
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
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
a(n) = 5^((n-1)/2)*ChebyshevU(n-1, 1/sqrt(5)). - G. C. Greubel, Jan 11 2024
Extensions
More terms from Paul Barry, Sep 20 2003
Comments