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A130255 Maximal index k of an odd Fibonacci number (A001519) such that A001519(k) = Fibonacci(2k-1) <= n (the 'lower' odd Fibonacci Inverse).

%I A130255 #17 Sep 08 2022 08:45:30
%S A130255 1,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,
%T A130255 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,
%U A130255 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6
%N A130255 Maximal index k of an odd Fibonacci number (A001519) such that A001519(k) = Fibonacci(2k-1) <= n (the 'lower' odd Fibonacci Inverse).
%C A130255 Inverse of the odd Fibonacci sequence (A001519), nearly, since a(A001519(n))=n except for n=0 (see A130256 for another version). a(n)+1 is the number of odd Fibonacci numbers (A001519) <= n (for n >= 1).
%H A130255 G. C. Greubel, <a href="/A130255/b130255.txt">Table of n, a(n) for n = 1..10000</a>
%F A130255 a(n) = floor((1 + arcsinh(sqrt(5)*n/2)/log(phi))/2).
%F A130255 a(n) = floor((1 + arccosh(sqrt(5)*n/2)/log(phi))/2).
%F A130255 a(n) = floor((1 + log_phi(sqrt(5)*n))/2) for n >= 1, where phi = (1 + sqrt(5))/2.
%F A130255 G.f.: g(x) = 1/(1-x)*Sum_{k>=1} x^Fibonacci(2k-1).
%F A130255 a(n) = floor((1/2)*(1 + log_phi(sqrt(5)*n + 1))) for n >= 1.
%e A130255 a(10)=3 because A001519(3) = 5 <= 10, but A001519(4) = 13 > 10.
%t A130255 Table[Floor[(1 +ArcSinh[Sqrt[5]*n/2]/Log[GoldenRatio])/2], {n, 1, 100}] (* _G. C. Greubel_, Sep 09 2018 *)
%o A130255 (PARI) phi=(1+sqrt(5))/2; vector(100, n, floor((1 +asinh(sqrt(5)*n/2)/log(phi))/2)) \\ _G. C. Greubel_, Sep 09 2018
%o A130255 (Magma) phi:=(1+Sqrt(5))/2; [Floor((1 +Argsinh(Sqrt(5)*n/2)/Log(phi))/2): n in [1..100]]; // _G. C. Greubel_, Sep 09 2018
%Y A130255 Cf. partial sums A130257. Other related sequences: A000045, A130233, A130237, A130239, A130256, A130259, A104160. Lucas inverse: A130241 - A130248.
%K A130255 nonn
%O A130255 1,2
%A A130255 _Hieronymus Fischer_, May 24 2007, Jul 02 2007