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 A130233 a(n) is the maximal k such that Fibonacci(k) <= n (the "lower" Fibonacci Inverse). 40
 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, 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, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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). LINKS Charles R Greathouse IV, Table of n, a(n) for n = 0..10000 FORMULA a(n) = floor(log_phi((sqrt(5)*n + sqrt(5*n^2+4))/2)) where phi = (1+sqrt(5))/2 = A001622. a(n) = floor(arcsinh(sqrt(5)*n/2) / log(phi)), with log(phi) = A002390. a(n) = A130234(n+1) - 1. G.f.: g(x) = 1/(1-x) * Sum_{k>=1} x^Fibonacci(k). a(n) = floor(log_phi(sqrt(5)*n+1)), n >= 0, where phi is the golden ratio. - Hieronymus Fischer, Jul 02 2007 EXAMPLE a(10) = 6, since Fibonacci(6) = 8 <= 10 but Fibonacci(7) = 13 > 10. MATHEMATICA 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 *) PROG (PARI) a(n)=log(sqrt(5)*n+1.5)\log((1+sqrt(5))/2) \\ Charles R Greathouse IV, Mar 21 2012 CROSSREFS Cf. A130235 (partial sums), A104162 (first differences). Other related sequences: A000045, A130234, A130237, A130239, A130255, A130259, A108852. Lucas inverse: A130241. Cf. A001622 (golden ratio), A002390 (its log). Sequence in context: A199769 A030601 A049839 * A131234 A349983 A204924 Adjacent sequences: A130230 A130231 A130232 * A130234 A130235 A130236 KEYWORD nonn,easy AUTHOR Hieronymus Fischer, May 17 2007 STATUS approved

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Last modified September 10 04:27 EDT 2024. Contains 375773 sequences. (Running on oeis4.)