OFFSET
0,4
COMMENTS
Transform of the Fibonacci numbers by the Chebyshev related transform which maps g(x) -> (1/(1-x^2))g(x/(1-x^2)).
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
D. Huylebrouck, The Meta-Golden Ratio Chi, Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture.
Index entries for linear recurrences with constant coefficients, signature (1,3,-1,-1).
FORMULA
G.f.: x/(1-x-3*x^2+x^3+x^4).
a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*F(n-2*k).
a(n) = Sum_{k=0..n} C((n+k)/2, k)*(1+(-1)^(n-k))*F(k)/2.
a(n) = (Fibonacci(n+1, (1+sqrt(5))/2) - Fibonacci(n+1, (1-sqrt(5))/2) )/sqrt(5), where Fibonacci(n,x) is the Fibonacci polynomial (see A011973). - G. C. Greubel, Jul 29 2019
MATHEMATICA
(* see A192232 for Mmca code. - M. F. Hasler, Apr 05 2016 *)
PROG
(PARI) Vec(x/(1-x-3*x^2+x^3+x^4)+O(x^40)) \\ M. F. Hasler, Apr 05 2016
(Magma) R<x>:=PowerSeriesRing(Integers(), 40); [0] cat Coefficients(R!( x/(1-x-3*x^2+x^3+x^4) )); // G. C. Greubel, Jul 29 2019
(Sage) (x/(1-x-3*x^2+x^3+x^4)).series(x, 40).coefficients(x, sparse=False) # G. C. Greubel, Jul 29 2019
(GAP) a:=[0, 1, 1, 4];; for n in [5..40] do a[n]:=a[n-1]+3*a[n-2]-a[n-3] -a[n-4]; od; a; # G. C. Greubel, Jul 29 2019
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Paul Barry, Sep 14 2005
STATUS
approved