cp's OEIS Frontend

A130256 Minimal index k of an odd Fibonacci number A001519 such that A001519(k) = Fibonacci(2*k-1) >= n (the 'upper' odd Fibonacci Inverse).

%I A130256 #20 Sep 08 2022 08:45:30
%S A130256 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 A130256 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 A130256 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 A130256 Minimal index k of an odd Fibonacci number A001519 such that A001519(k) = Fibonacci(2*k-1) >= n (the 'upper' odd Fibonacci Inverse).
%C A130256 Inverse of the odd Fibonacci sequence (A001519), nearly, since a(A001519(n))=n except for n=1 (see A130255 for another version).
%C A130256 a(n+1) is the number of odd Fibonacci numbers (A001519) <= n (for n >= 0).
%H A130256 G. C. Greubel, <a href="/A130256/b130256.txt">Table of n, a(n) for n = 0..10000</a>
%F A130256 a(n) = ceiling((1+arccosh(sqrt(5)*n/2)/log(phi))/2), where phi=(1+sqrt(5))/2.
%F A130256 G.f.: (x/(1-x))*Sum_{k>=0} x^Fibonacci(2*k-1).
%F A130256 a(n) = ceiling((1/2)*(1+log_phi(sqrt(5)*n-1))) for n >= 2, where phi=(1+sqrt(5))/2.
%e A130256 a(10)=4 because A001519(4) = 13 >= 10, but A001519(3) = 5 < 10.
%t A130256 Join[{0, 0}, Table[Ceiling[1/2*(1 + Log[GoldenRatio, (Sqrt[5]*n - 1)])], {n, 2, 100}]] (* _G. C. Greubel_, Sep 12 2018 *)
%o A130256 (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 A130256 (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 A130256 Cf. partial sums A130258.
%Y A130256 Other related sequences: A000045, A001906, A130234, A130237, A130239, A130255, A130260.
%Y A130256 Lucas inverse: A130241 - A130248.
%K A130256 nonn
%O A130256 0,3
%A A130256 _Hieronymus Fischer_, May 24 2007, Jul 02 2007