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%I #31 Mar 15 2022 03:02:04
%S 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,
%T 9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,
%U 10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10
%N a(n) is the maximal k such that Fibonacci(k) <= n (the "lower" Fibonacci Inverse).
%C Inverse of the Fibonacci sequence (A000045), nearly, since a(Fibonacci(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).
%H Charles R Greathouse IV, <a href="/A130233/b130233.txt">Table of n, a(n) for n = 0..10000</a>
%F a(n) = floor(log_phi((sqrt(5)*n + sqrt(5*n^2+4))/2)) where phi = (1+sqrt(5))/2 = A001622.
%F a(n) = floor(arcsinh(sqrt(5)*n/2) / log(phi)), with log(phi) = A002390.
%F a(n) = A130234(n+1) - 1.
%F G.f.: g(x) = 1/(1-x) * Sum_{k>=1} x^Fibonacci(k).
%F a(n) = floor(log_phi(sqrt(5)*n+1)), n >= 0, where phi is the golden ratio. - _Hieronymus Fischer_, Jul 02 2007
%e a(10) = 6, since Fibonacci(6) = 8 <= 10 but Fibonacci(7) = 13 > 10.
%t fibLLog[0] := 0; fibLLog[1] := 2; fibLLog[n_Integer] := fibLLog[n] = If[n < Fibonacci[fibLLog[n - 1] + 1], fibLLog[n - 1], fibLLog[n - 1] + 1]; Table[fibLLog[n], {n, 0, 88}] (* _Alonso del Arte_, Sep 01 2013 *)
%o (PARI) a(n)=log(sqrt(5)*n+1.5)\log((1+sqrt(5))/2) \\ _Charles R Greathouse IV_, Mar 21 2012
%Y Cf. A130235 (partial sums), A104162 (first differences).
%Y Other related sequences: A000045, A130234, A130237, A130239, A130255, A130259, A108852. Lucas inverse: A130241.
%Y Cf. A001622 (golden ratio), A002390 (its log).
%K nonn,easy
%O 0,2
%A _Hieronymus Fischer_, May 17 2007
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