OFFSET
0,3
LINKS
Michael De Vlieger, Table of n, a(n) for n = 0..2096
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).
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
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 *)
PROG
(PARI) my(x='x+O('x^30)); concat([0], Vec(x/(1-4*x+9*x^2))) \\ 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
(Sage)
def A190967_list(prec):
P.<x> = PowerSeriesRing(ZZ, prec)
return P( x/(1-4*x+9*x^2) ).list()
A190967_list(30) # G. C. Greubel, Dec 07 2019
(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
CROSSREFS
KEYWORD
sign,easy
AUTHOR
Vladimir Joseph Stephan Orlovsky, May 24 2011
STATUS
approved