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A108852 Number of Fibonacci numbers <= n.

%I A108852 #39 Nov 04 2024 16:31:10
%S A108852 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 A108852 10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,
%U A108852 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 A108852 Number of Fibonacci numbers <= n.
%C A108852 1 is counted twice as a Fibonacci number: F(1) = F(2) = 1. - _Alois P. Heinz_, Nov 04 2024
%H A108852 Michael De Vlieger, <a href="/A108852/b108852.txt">Table of n, a(n) for n = 0..10000</a>
%H A108852 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 A108852 G.f.: (Sum_{n>=0} x^Fibonacci(n))/(1-x). - _Vladeta Jovovic_, Nov 27 2005
%F A108852 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 A108852 a(n) = 1+floor(log_phi(sqrt(5)*n+1)), n>=0, where phi is the golden ratio. - _Hieronymus Fischer_, Jul 02 2007
%p A108852 a:= proc(n) option remember; `if`(n<2, 2*n+1, a(n-1)+
%p A108852       (t-> `if`(issqr(t+4) or issqr(t-4), 1, 0))(5*n^2))
%p A108852     end:
%p A108852 seq(a(n), n=0..100);  # _Alois P. Heinz_, Nov 04 2024
%t A108852 fibPi[n_] := 1 + Floor[ Log[ GoldenRatio, 1 + n*Sqrt@ 5]]; Array[fibPi, 80, 0] (* _Robert G. Wilson v_, Aug 03 2014 *)
%o A108852 (Haskell) fibs :: [Integer]
%o A108852 fibs = 0 : 1 : zipWith (+) fibs (tail fibs)
%o A108852 fibs_to :: Integer -> Integer
%o A108852 fibs_to n = length $ takeWhile (<= n) fibs
%o A108852 (Python)
%o A108852 def A108852(n):
%o A108852     a, b, c = 0, 1, 0
%o A108852     while a <= n:
%o A108852         a, b = b, a+b
%o A108852         c += 1
%o A108852     return c # _Chai Wah Wu_, Nov 04 2024
%Y A108852 Cf. A000045, A001622, A060384, A072649.
%Y A108852 Partial sums of A104162.
%K A108852 nonn
%O A108852 0,2
%A A108852 Michael C. Vanier (mvanier(AT)cs.caltech.edu), Nov 27 2005