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An inverse Catalan transform of Fibonacci(2n).
7

%I #32 Jan 30 2023 02:37:30

%S 0,1,2,2,0,-5,-13,-21,-21,0,55,144,233,233,0,-610,-1597,-2584,-2584,0,

%T 6765,17711,28657,28657,0,-75025,-196418,-317811,-317811,0,832040,

%U 2178309,3524578,3524578,0,-9227465,-24157817,-39088169,-39088169,0,102334155,267914296,433494437,433494437,0,-1134903170

%N An inverse Catalan transform of Fibonacci(2n).

%H G. C. Greubel, <a href="/A100334/b100334.txt">Table of n, a(n) for n = 0..1000</a>

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,2,-1).

%F G.f.: x*(1-x)/(1-3*x+4*x^2-2*x^3+x^4).

%F a(n) = (phi)^n*sqrt(2/5+2*sqrt(5)/25)*sin(Pi*(n+1)/5) -(1/phi)^n*sqrt(2/5-2*sqrt(5)/25)*sin(2*Pi*(n+1)/5), where phi=(1+sqrt(5))/2;

%F a(n) = Sum_{k=0..floor(n/2)} (C(n-k, k)*(-1)^k*Sum_{j=0..n-k} C(n-k, j)*F(j));

%F a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*Fibonacci(2n-2k).

%F a(n) = 3*a(n-1)-4*a(n-2)+2*a(n-3)-a(n-4). - _Paul Curtz_, May 13 2008

%F a(n) = Sum_{k=0..n} A109466(n,k)*A001906(k). - _Philippe Deléham_, Oct 30 2008

%F a(5*n) = -F(-5*n), a(5*n+1) = -F(-5*n-2), a(5*n+2) = a(5*n+3) = F(-5*n-3), a(5*n+4) = 0. - _Ehren Metcalfe_, Apr 04 2019

%t Table[FullSimplify[GoldenRatio^n*Sqrt[2/5 + 2*Sqrt[5]/25]*Sin[Pi*n/5 + Pi/5] - (1/GoldenRatio)^n*Sqrt[2/5 - 2*Sqrt[5]/25]*Sin[2*Pi*n/5 + 2*Pi/5]], {n, 0, 41}] (* _Arkadiusz Wesolowski_, Oct 26 2012 *)

%t LinearRecurrence[{3,-4,2,-1}, {0,1,2,2}, 41] (* _G. C. Greubel_, Jan 30 2023 *)

%o (Magma) I:=[0,1,2,2]; [n le 4 select I[n] else 3*Self(n-1) -4*Self(n-2) +2*Self(n-3) -Self(n-4): n in [1..41]]; // _G. C. Greubel_, Jan 30 2023

%o (SageMath)

%o def A100334(n): return sum((-1)^k*binomial(n-k,k)*fibonacci(2*n-2*k) for k in range(1+(n//2)))

%o [A100334(n) for n in range(41)] # _G. C. Greubel_, Jan 30 2023

%Y Cf. A001906, A109466.

%Y Cf. A102312 (F(5n)), A134489 (F(5n+2)), A134490 (F(5n+3)).

%K easy,sign

%O 0,3

%A _Paul Barry_, Nov 17 2004