%I A130233
%S A130233 0,2,3,4,4,5,5,5,6,6,6,6,6,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,9,
%T A130233 9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,
%U A130233 10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10
%N A130233 Maximal index k of a Fibonacci number such that Fib(k)<=n (the 'lower' 
               Fibonacci Inverse).
%C A130233 Inverse of the Fibonacci sequence (A000045), nearly, since a(Fib(n))=n 
               except for n=1 (see A130234 for another version). a(n)+1 is equal 
               to the partial sum of the Fibonacci indicator sequence (see A104162).
%F A130233 a(n)=floor(log_phi((sqr(5)*n+sqr(5*n^2+4))/2))=floor(arsinh(sqr(5)*n/
               2)/ln(phi)) where phi=(1+sqr(5))/2. Also true: a(n)=A130234(n+1)-1. 
               G.f.: g(x)=1/(1-x)*sum{k>=1, x^Fib(k)}.
%F A130233 a(n)=floor(log_phi(sqr(5)*n+1)), n>=0, where phi is the = golden ratio. 
               - Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), Jul 02 2007
%e A130233 a(10)=6, since Fib(6)=8<=10 but Fib(7)=13>10.
%Y A130233 Partial sums: A130235. Other related sequences: A000045, A130234, A130237, 
               A130239, A130255, A130259, A104162, A108852, A130255, A130259. Lucas 
               inverse: A130241 - A130248.
%Y A130233 Sequence in context: A092338 A030601 A049839 this_sequence A131234 A056791 
               A027434
%Y A130233 Adjacent sequences: A130230 A130231 A130232 this_sequence A130234 A130235 
               A130236
%K A130233 nonn
%O A130233 0,2
%A A130233 Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), May 17 2007