This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A100334 #32 Jan 30 2023 02:37:30
%S A100334 0,1,2,2,0,-5,-13,-21,-21,0,55,144,233,233,0,-610,-1597,-2584,-2584,0,
%T A100334 6765,17711,28657,28657,0,-75025,-196418,-317811,-317811,0,832040,
%U A100334 2178309,3524578,3524578,0,-9227465,-24157817,-39088169,-39088169,0,102334155,267914296,433494437,433494437,0,-1134903170
%N A100334 An inverse Catalan transform of Fibonacci(2n).
%H A100334 G. C. Greubel, <a href="/A100334/b100334.txt">Table of n, a(n) for n = 0..1000</a>
%H A100334 <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (3,-4,2,-1).
%F A100334 G.f.: x*(1-x)/(1-3*x+4*x^2-2*x^3+x^4).
%F A100334 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 A100334 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 A100334 a(n) = Sum_{k=0..floor(n/2)} C(n-k, k)*(-1)^k*Fibonacci(2n-2k).
%F A100334 a(n) = 3*a(n-1)-4*a(n-2)+2*a(n-3)-a(n-4). - _Paul Curtz_, May 13 2008
%F A100334 a(n) = Sum_{k=0..n} A109466(n,k)*A001906(k). - _Philippe Deléham_, Oct 30 2008
%F A100334 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 A100334 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 A100334 LinearRecurrence[{3,-4,2,-1}, {0,1,2,2}, 41] (* _G. C. Greubel_, Jan 30 2023 *)
%o A100334 (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 A100334 (SageMath)
%o A100334 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 [A100334(n) for n in range(41)] # _G. C. Greubel_, Jan 30 2023
%Y A100334 Cf. A001906, A109466.
%Y A100334 Cf. A102312 (F(5n)), A134489 (F(5n+2)), A134490 (F(5n+3)).
%K A100334 easy,sign
%O A100334 0,3
%A A100334 _Paul Barry_, Nov 17 2004