A190967 a(n) = 4*a(n-1) - 9*a(n-2), with a(0)=0, a(1)=1.

0, 1, 4, 7, -8, -95, -308, -377, 1264, 8449, 22420, 13639, -147224, -711647, -1521572, 318535, 14968288, 57006337, 93310756, -139814009, -1399052840, -4337885279, -4760065556, 20000705287, 122843411152, 311367297025, 139878487732, -2242791722297

Offset: 0

Author

Vladimir Joseph Stephan Orlovsky, May 24 2011

Keywords

• sign
• easy

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..2096
• Mei Bai, Wenchang Chu, and Dongwei Guo, Reciprocal Formulae among Pell and Lucas Polynomials, Mathematics, 10, 2691, (2022). See p. 5.
• Beata Bajorska-Harapińska, Barbara Smoleń, and Roman Wituła, On Quaternion Equivalents for Quasi-Fibonacci Numbers, Shortly Quaternaccis, Advances in Applied Clifford Algebras (2019) Vol. 29, 54.
• Index entries for linear recurrences with constant coefficients, signature (4,-9).

Crossrefs

Cf. A190958 (index to generalized Fibonacci sequences).

Programs

GAP

a:=[0,1];; for n in [3..30] do a[n]:=4*a[n-1]-9*a[n-2]; od; a; # G. C. Greubel, Dec 07 2019

Magma

I:=[0,1]; [n le 2 select I[n] else 4*Self(n-1) - 9*Self(n-2): n in [1..30]]; // G. C. Greubel, Dec 07 2019

Maple

seq(coeff(series(x/(1-4*x+9*x^2), x, n+1), x, n), n = 0..30); # G. C. Greubel, Dec 07 2019

Mathematica

LinearRecurrence[{4,-9}, {0,1}, 50]
Table[FullSimplify[(3*I)^(n-1)*Fibonacci[n, -4*I/3]], {n,0,30}] (* G. C. Greubel, Dec 07 2019 *)

PARI

my(x='x+O('x^30)); concat([0], Vec(x/(1-4*x+9*x^2))) \\ G. C. Greubel, Dec 07 2019

Sage

def A190967_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x/(1-4*x+9*x^2) ).list()
A190967_list(30) # G. C. Greubel, Dec 07 2019

Formula

G.f.: x/(1-4*x+9*x^2). - Philippe Deléham, Oct 12 2011

a(n) = (3*i)^(n-1)*Fibonacci(n, -4*i/3), where i=sqrt(-1) and F(n,x) is the Fibonacci polynomial. - G. C. Greubel, Dec 07 2019

a(n) = (3^n*sin(n*arccos(2/3)))/sqrt(5) = 3^(n-1)*chebyshevU(n-1, 2/3). - Federico Provvedi, Feb 23 2022

E.g.f.: exp(2*x)*sin(sqrt(5)*x)/sqrt(5). - Stefano Spezia, Jul 24 2025