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

Author

N. J. A. Sloane

Keywords

• sign
• easy

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

• 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

Crossrefs

Cf. A006495, A006496, A084102, A088136, A088137, A088139, A094423.

Programs

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

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 *)

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

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