A108852 Number of Fibonacci numbers <= n.

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

Offset: 0

Author

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

Keywords

• nonn

Comments

1 is counted twice as a Fibonacci number: F(1) = F(2) = 1. - Alois P. Heinz, Nov 04 2024

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, 17-36.

Crossrefs

Cf. A000045, A001622, A060384, A072649.

Partial sums of A104162.

Programs

Haskell

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

Maple

a:= proc(n) option remember; `if`(n<2, 2*n+1, a(n-1)+
      (t-> `if`(issqr(t+4) or issqr(t-4), 1, 0))(5*n^2))
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Nov 04 2024

Mathematica

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

Python

def A108852(n):
    a, b, c = 0, 1, 0
    while a <= n:
        a, b = b, a+b
        c += 1
    return c # Chai Wah Wu, Nov 04 2024

Formula

G.f.: (Sum_{n>=0} x^Fibonacci(n))/(1-x). - 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