%I #20 Sep 08 2022 08:45:30

%S 0,0,2,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,

%T 5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,

%U 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7

%N Minimal index k of an odd Fibonacci number A001519 such that A001519(k) = Fibonacci(2*k-1) >= n (the 'upper' odd Fibonacci Inverse).

%C Inverse of the odd Fibonacci sequence (A001519), nearly, since a(A001519(n))=n except for n=1 (see A130255 for another version).

%C a(n+1) is the number of odd Fibonacci numbers (A001519) <= n (for n >= 0).

%H G. C. Greubel, <a href="/A130256/b130256.txt">Table of n, a(n) for n = 0..10000</a>

%F a(n) = ceiling((1+arccosh(sqrt(5)*n/2)/log(phi))/2), where phi=(1+sqrt(5))/2.

%F G.f.: (x/(1-x))*Sum_{k>=0} x^Fibonacci(2*k-1).

%F a(n) = ceiling((1/2)*(1+log_phi(sqrt(5)*n-1))) for n >= 2, where phi=(1+sqrt(5))/2.

%e a(10)=4 because A001519(4) = 13 >= 10, but A001519(3) = 5 < 10.

%t Join[{0, 0}, Table[Ceiling[1/2*(1 + Log[GoldenRatio, (Sqrt[5]*n - 1)])], {n, 2, 100}]] (* _G. C. Greubel_, Sep 12 2018 *)

%o (PARI) for(n=0,100, print1(if(n==0, 0, if(n==1, 0, ceil((1/2)*(1 + log(sqrt(5)*n-1)/(log((1+sqrt(5))/2)))))), ", ")) \\ _G. C. Greubel_, Sep 12 2018

%o (Magma) [0,0] cat [Ceiling((1/2)*(1 + Log(Sqrt(5)*n-1)/(Log((1+Sqrt(5))/2)))): n in [2..100]]; // _G. C. Greubel_, Sep 12 2018

%Y Cf. partial sums A130258.

%Y Other related sequences: A000045, A001906, A130234, A130237, A130239, A130255, A130260.

%Y Lucas inverse: A130241 - A130248.

%K nonn

%O 0,3

%A _Hieronymus Fischer_, May 24 2007, Jul 02 2007