|
%I #25 Jun 11 2021 18:25:00
%S 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,
%T 10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,
%U 11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11
%N Number of Fibonacci numbers <= n.
%H Michael De Vlieger, <a href="/A108852/b108852.txt">Table of n, a(n) for n = 0..10000</a>
%H Dorin Andrica, Ovidiu Bagdasar, and George Cătălin Tųrcąs, <a href="https://doi.org/10.2478/auom-2021-0002">On some new results for the generalised Lucas sequences</a>, An. Şt. Univ. Ovidius Constanţa (Romania, 2021) Vol. 29, No. 1, 17-36.
%F G.f.: (Sum_{n>=0} x^Fibonacci(n))/(1-x). - _Vladeta Jovovic_, Nov 27 2005
%F 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
%F a(n) = 1+floor(log_phi(sqrt(5)*n+1)), n>=0, where phi is the golden ratio. - _Hieronymus Fischer_, Jul 02 2007
%t fibPi[n_] := 1 + Floor[ Log[ GoldenRatio, 1 + n*Sqrt@ 5]]; Array[fibPi, 80, 0] (* _Robert G. Wilson v_, Aug 03 2014 *)
%o (Haskell) fibs :: [Integer]
%o fibs = 0 : 1 : zipWith (+) fibs (tail fibs)
%o fibs_to :: Integer -> Integer
%o fibs_to n = length $ takeWhile (<= n) fibs
%Y Cf. A060384, A072649.
%K nonn
%O 0,2
%A Michael C. Vanier (mvanier(AT)cs.caltech.edu), Nov 27 2005
|
