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%I #16 Sep 08 2022 08:45:30
%S 0,1,1,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,
%T 4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,
%U 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5
%N Maximal index k of an even Fibonacci number (A001906) such that A001906(k) = Fib(2k) <= n (the 'lower' even Fibonacci Inverse).
%C Inverse of the even Fibonacci sequence (A001906), since a(A001906(n))=n (see A130260 for another version).
%C a(n)+1 is the number of even Fibonacci numbers (A001906) <=n.
%H G. C. Greubel, <a href="/A130259/b130259.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) = floor(arcsinh(sqrt(5)*n/2)/(2*log(phi))), where phi=(1+sqrt(5))/2.
%F a(n) = A130260(n+1) - 1.
%F G.f.: g(x) = 1/(1-x)*Sum_{k>=1} x^Fibonacci(2*k).
%F a(n) = floor(1/2*log_phi(sqrt(5)*n+1)) for n>=0.
%e a(10)=3 because A001906(3)=8<=10, but A001906(4)=21>10.
%t Table[Floor[1/2*Log[GoldenRatio, (Sqrt[5]*n + 1)]], {n, 0, 100}] (* _G. C. Greubel_, Sep 12 2018 *)
%o (PARI) vector(100, n, n--; floor(log((sqrt(5)*n+1))/(2*log((1+sqrt(5))/2) ))) \\ _G. C. Greubel_, Sep 12 2018
%o (Magma) [Floor(Log((Sqrt(5)*n+1))/(2*Log((1+Sqrt(5))/2)))): n in [0..100]]; // _G. C. Greubel_, Sep 12 2018
%Y Cf. partial sums A130261. Other related sequences: A000045, A001519, A130233, A130237, A130239, A130255, A130260, A104160. Lucas inverse: A130241 - A130248.
%K nonn
%O 0,4
%A _Hieronymus Fischer_, May 25 2007, Jul 02 2007
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