%I A130247
%S A130247 1,0,2,3,3,3,4,4,4,4,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,
%T A130247 7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,8,
%U A130247 8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9
%N A130247 Inverse Lucas (A000032) numbers: index k of a Lucas number such that 
               Lucas(k)=n; max(k|Lucas(k)<n), if there is no such index.
%C A130247 Inverse of the Lucas sequence (A000032), since a(Lucas(n))=n for n>=0 
               (see A130241 and A130242 for other versions). Same as A130241 except 
               for n=1.
%F A130247 a(n)=c(n), if (n^2-4)/5 is a square number, a(n)=s(n), if (n^2+4)/5 is 
               a square number and a(n)=floor(log_phi(n)) else, where s(n)=floor(arsinh(n/
               2)/ln(phi)), c(n)=floor(arcosh(n/2)/ln(phi)) and phi=(1+sqr(5))/2.
%F A130247 a(n)=A130241(n) except for n=2.
%F A130247 G.f.: g(x)=1/(1-x)*sum{k>=1, x^Lucas(k)}-x^2.
%F A130247 a(n)=floor(log_phi(n+1/2)) for n>=3, where phi is the golden ratio.
%e A130247 a(2)=0, since Lucas(0)=2; a(10)=4, since Lucas(4)=7<10 but Lucas(5)=11>
               10.
%Y A130247 For partial sums see A130248. Other related sequences: A000032, A130241, 
               A130242, A130245, A130249, A130255, A130259. Indicator sequence A102460. 
               Fibonacci inverse see A130233 - A130240, A104162.
%Y A130247 Sequence in context: A117806 A085423 A130241 this_sequence A087839 A106742 
               A106733
%Y A130247 Adjacent sequences: A130244 A130245 A130246 this_sequence A130248 A130249 
               A130250
%K A130247 nonn
%O A130247 1,3
%A A130247 Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), May 19 2007, Jul 02 
               2007