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Maximal index k of an even Fibonacci number (A001906) such that A001906(k) = Fib(2k) <= n (the 'lower' even Fibonacci Inverse).
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%I #20 Sep 28 2024 14:23:08

%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