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%I #18 Sep 08 2022 08:45:22
%S 0,1,1,4,6,16,29,67,132,288,588,1253,2597,5480,11430,24020,50233,
%T 105383,220632,462528,968808,2030377,4253641,8913436,18675174,
%U 39131464,81989909,171795691,359958780,754224480,1580315220,3311234189
%N A Chebyshev-related transform of the Fibonacci numbers.
%C Transform of the Fibonacci numbers by the Chebyshev related transform which maps g(x) -> (1/(1-x^2))g(x/(1-x^2)).
%H G. C. Greubel, <a href="/A112576/b112576.txt">Table of n, a(n) for n = 0..1000</a>
%H D. Huylebrouck, <a href="http://archive.bridgesmathart.org/2014/bridges2014-151.pdf">The Meta-Golden Ratio Chi</a>, Proceedings of Bridges 2014: Mathematics, Music, Art, Architecture, Culture.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (1,3,-1,-1).
%F G.f.: x/(1-x-3*x^2+x^3+x^4).
%F a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*F(n-2*k).
%F a(n) = Sum_{k=0..n} C((n+k)/2, k)*(1+(-1)^(n-k))*F(k)/2.
%F 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
%t (* see A192232 for Mmca code. - _M. F. Hasler_, Apr 05 2016 *)
%o (PARI) Vec(x/(1-x-3*x^2+x^3+x^4)+O(x^40)) \\ _M. F. Hasler_, Apr 05 2016
%o (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
%o (Sage) (x/(1-x-3*x^2+x^3+x^4)).series(x, 40).coefficients(x, sparse=False) # _G. C. Greubel_, Jul 29 2019
%o (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
%Y Cf. A011973, A192232.
%K easy,nonn
%O 0,4
%A _Paul Barry_, Sep 14 2005
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