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A045873 a(n) = A006496(n)/2. 13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Partial sums of A006495. - Paul Barry, Mar 16 2006
This is the Lucas U(P=2,Q=5) sequence. - R. J. Mathar, Oct 24 2012
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
LINKS
Ronald Orozco López, Deformed Differential Calculus on Generalized Fibonacci Polynomials, arXiv:2211.04450 [math.CO], 2022.
Wikipedia, Lucas sequence
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
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
MATHEMATICA
LinearRecurrence[{2, -5}, {0, 1}, 40] (* G. C. Greubel, Jan 11 2024 *)
PROG
(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
(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
(PARI) concat(0, Vec(1/(1-2*x+5*x^2)+O(x^99))) \\ Charles R Greathouse IV, Dec 22 2011
(Sage) [lucas_number1(n, 2, 5) for n in range(0, 29)] # Zerinvary Lajos, Apr 23 2009
(SageMath)
A045873=BinaryRecurrenceSequence(2, -5, 0, 1)
[A045873(n) for n in range(41)] # G. C. Greubel, Jan 11 2024
CROSSREFS
Sequence in context: A151508 A164826 A055392 * A265022 A110060 A061081
KEYWORD
sign,easy
AUTHOR
EXTENSIONS
More terms from Paul Barry, Sep 20 2003
STATUS
approved

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Last modified February 23 11:40 EST 2024. Contains 370283 sequences. (Running on oeis4.)