

A108852


Number of Fibonacci numbers <= n.


18



1, 3, 4, 5, 5, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 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, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11
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OFFSET

0,2


LINKS



FORMULA

a(n) = 1+floor(log_phi((sqrt(5)*n+sqrt(5*n^2+4))/2)), n>=0, where phi is the golden ratio. Alternatively, a(n)=1+floor(arcsinh(sqrt(5)*n/2)/log(phi)). Also a(n)=A072649(n)+2.  Hieronymus Fischer, May 02 2007
a(n) = 1+floor(log_phi(sqrt(5)*n+1)), n>=0, where phi is the golden ratio.  Hieronymus Fischer, Jul 02 2007


MATHEMATICA

fibPi[n_] := 1 + Floor[ Log[ GoldenRatio, 1 + n*Sqrt@ 5]]; Array[fibPi, 80, 0] (* Robert G. Wilson v, Aug 03 2014 *)


PROG

(Haskell) fibs :: [Integer]
fibs = 0 : 1 : zipWith (+) fibs (tail fibs)
fibs_to :: Integer > Integer
fibs_to n = length $ takeWhile (<= n) fibs


CROSSREFS



KEYWORD

nonn


AUTHOR

Michael C. Vanier (mvanier(AT)cs.caltech.edu), Nov 27 2005


STATUS

approved



