A122690 a(n) = 5*a(n-1) + 4*a(n-2) with a(0)=1, a(1)=4.

1, 4, 24, 136, 776, 4424, 25224, 143816, 819976, 4675144, 26655624, 151978696, 866515976, 4940494664, 28168537224, 160604664776, 915697472776, 5220906022984, 29767320006024, 169720224122056, 967670400634376

Offset: 0

Author

Philippe Deléham, Sep 22 2006

Keywords

• nonn
• easy
• less

Links

• Harvey P. Dale, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (5,4).

Crossrefs

Cf. A015537.

Programs

GAP

a:=[1,4];; for n in [3..35] do a[n]:=5*a[n-1]+4*a[n-2]; od; a; # G. C. Greubel, Dec 26 2019

Magma

R:=PowerSeriesRing(Integers(), 35); Coefficients(R!( (1-x)/(1-5*x-4*x^2) )); // G. C. Greubel, Dec 26 2019

Maple

seq(coeff(series((1-x)/(1-5*x-4*x^2), x, n+1), x, n), n = 0..35); # G. C. Greubel, Dec 26 2019

Mathematica

LinearRecurrence[{5,4},{1,4},35] (* Harvey P. Dale, Apr 06 2012 *)
Table[2^(n-1)*(2*Fibonacci[n+1, 5/2] - Fibonacci[n, 5/2]), {n,0,35}] (* G. C. Greubel, Dec 26 2019 *)

PARI

Vec((1-x)/(1-5*x-4*x^2)+O(x^35)) \\ Charles R Greathouse IV, Jan 17 2012

Sage

def A122690_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-x)/(1-5*x-4*x^2) ).list()
A122690_list(35) # G. C. Greubel, Dec 26 2019

Formula

a(n) = Sum_{k=0..n} 4^k*A122542(n,k).

G.f.: (1-x)/(1-5*x-4*x^2).

Lim_{n -> infinity} a(n+1)/a(n)-> (5+sqrt(41))/2 = 5.701562118716.

a(n) = 2^(n-1)*(2*Fibonacci(n+1, 5/2) - Fibonacci(n, 5/2)) = -(2/I)^(n-1)*( 2*I*ChebyshevU(n, 5*I/4) + ChebyshevU(n-1, 5*I/4)). - G. C. Greubel, Dec 26 2019

Extensions

Corrected by T. D. Noe, Nov 07 2006