

A108852


Number of Fibonacci numbers <= n.


17



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

Michael De Vlieger, Table of n, a(n) for n = 0..10000
Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, On some new results for the generalised Lucas sequences, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 1736.


FORMULA

G.f.: (Sum_{n>=0} x^Fibonacci(n))/(1x).  Vladeta Jovovic, Nov 27 2005
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

Cf. A060384, A072649.
Sequence in context: A098200 A092405 A130234 * A179413 A119476 A358700
Adjacent sequences: A108849 A108850 A108851 * A108853 A108854 A108855


KEYWORD

nonn


AUTHOR

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


STATUS

approved



