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%I #32 Sep 08 2022 08:45:13
%S 1,2,5,15,50,175,625,2250,8125,29375,106250,384375,1390625,5031250,
%T 18203125,65859375,238281250,862109375,3119140625,11285156250,
%U 40830078125,147724609375,534472656250,1933740234375,6996337890625
%N Binomial transform of Fibonacci(2n-1) (A001519).
%H G. C. Greubel, <a href="/A093129/b093129.txt">Table of n, a(n) for n = 0..1000</a>
%H <a href="/index/Rec#order_02">Index entries for linear recurrences with constant coefficients</a>, signature (5,-5).
%F G.f.: (1-3*x)/(1-5*x+5*x^2).
%F a(n) = (5-sqrt(5))*((5+sqrt(5))/2)^n/10 + (5+sqrt(5))*((5-sqrt(5))/2)^n/10.
%F a(n) = A093123(n)/2^n.
%F a(n) = A020876(n-1). - _R. J. Mathar_, Sep 05 2008
%F a(n) = A030191(n) - 3*A030191(n-1). - _R. J. Mathar_, Jun 29 2012
%F a(2*n) = 5^n*Fibonacci(2*n-1), a(2*n+1) = 5^n*Lucas(2*n). - _G. C. Greubel_, Dec 27 2019
%F E.g.f.: (1/10)*exp((1/2)*(5-sqrt(5))*x)*(5 + sqrt(5) + (5 - sqrt(5))*exp(sqrt(5)*x)). - _Stefano Spezia_, Dec 28 2019
%p a:= n-> (<<0|1>, <-5|5>>^n. <<1,2>>)[1,1]:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Aug 29 2015
%t LinearRecurrence[{5, -5}, {1, 2}, 25] (* _Jean-François Alcover_, May 11 2019 *)
%t Table[If[EvenQ[n], 5^(n/2)*Fibonacci[n-1], 5^((n-1)/2)*LucasL[n-1]], {n,0,30}] (* _G. C. Greubel_, Dec 27 2019 *)
%o (Sage) [lucas_number2(n,5,5) for n in range(-1,25)] # _Zerinvary Lajos_, Jul 08 2008
%o (PARI) my(x='x+O('x^30)); Vec((1-3*x)/(1-5*x+5*x^2)) \\ _G. C. Greubel_, Dec 27 2019
%o (Magma) I:=[1,2]; [n le 2 select I[n] else 5*(Self(n-1) - Self(n-2)): n in [1..30]]; // _G. C. Greubel_, Dec 27 2019
%o (GAP) a:=[1,2];; for n in [3..30] do a[n]:=5*(a[n-1]-a[n-2]); od; a; # _G. C. Greubel_, Dec 27 2019
%Y Cf. A000032, A000045, A001519, A020876, A030191, A093123.
%K easy,nonn
%O 0,2
%A _Paul Barry_, Mar 23 2004
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