A020909 Number of bits in the base-2 representation of the n-th Fibonacci number.

1, 1, 2, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 10, 10, 11, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 21, 22, 22, 23, 24, 24, 25, 26, 26, 27, 28, 28, 29, 30, 31, 31, 32, 33, 33, 34, 35, 35, 36, 37, 38, 38, 39, 40, 40, 41, 42, 42, 43, 44, 44, 45, 46, 47, 47, 48, 49, 49, 50, 51

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy
• base

Crossrefs

Cf. A000045, A070939.

Programs

Mathematica

f[n_] := 1 + Floor@ Log2@ Fibonacci@ n; Array[f, 74] (* Robert G. Wilson v, Jun 24 2011 *)
IntegerLength[#,2]&/@Fibonacci[Range[80]] (* Harvey P. Dale, Feb 11 2015 *)

PARI

a(n)={
  if(n<99,return(#binary(fibonacci(n))));
  localbitprec(logint(n*6351109\9148265,2)+64);
  my(s=sqrt(5),phi=(s+1)/2,x=n*log(phi)-log(s));
  ceil(x/log(2));
} \\ Charles R Greathouse IV, Aug 24 2021

Formula

a(n) = 1 + floor( log_2( A000045(n) ) ).

a(n) = A070939(A000045(n)). - R. J. Mathar, Aug 11 2012

Extensions

More terms from James Sellers