A050815 Number of positive Fibonacci numbers with n decimal digits.

6, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 4, 5, 5, 5, 5

Offset: 1

Author

Patrick De Geest, Oct 15 1999

Keywords

• nonn
• base

Comments

If n>1 then a(n) = 4 or 5. - Robert Gerbicz, Sep 05 2002

The sequence is almost periodic, see also A072353. - Reinhard Zumkeller, Apr 14 2005

Examples

			At length 1 there are 6 such numbers: 1, 1, 2, 3, 5 and 8.
		

Links

• Hans J. H. Tuenter, Table of n, a(n) for n = 1..1001
• Andreas Guthmann, Wieviele k-stellige Fibonaccizahlen gibt es?, Archiv der Mathematik, Vol. 59, No. 4 (1992), pp. 334-340.
• Ron Knott, The Fibonacci Numbers.
• Ron Knott, Fibonacci Numbers and the Golden Section.
• Ron Knott, [Number of digits in Fib(i)] : Calculator.
• Jan-Christoph Puchta, The Number of k-Digit Fibonacci Numbers, The Fibonacci Quarterly, Vol. 39, No. 4 (2001), pp. 334-335.
• Jürgen Spilker, Die Ziffern der Fibonacci-Zahlen, Elemente der Mathematik, Vol. 58 (Birkhäuser 2003), pp. 26-33.
• Eric Weisstein's World of Mathematics, Fibonacci Number.
• Eric Weisstein's World of Mathematics, Almost Periodic Function.

Crossrefs

See A098842 for another version.

Cf. A000045, A001622, A002390, A072354, A105563, A105565, A060384, A097348.

Programs

Mathematica

Drop[Last/@Tally[Table[IntegerLength[Fibonacci[n]],{n,505}]],-1] (* Jayanta Basu, Jun 01 2013 *)

Formula

Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = log(10)/log(phi) = 1/A097348 = 4.7849719667... - Amiram Eldar, Jan 12 2022

For n>1, a(n) = 4+[{n*alpha+beta}<{alpha}], where alpha=log(10)/log(phi), beta=log(5)/(2*log(phi)), [X] is the Iverson bracket, {x}=x-floor(x), denotes the fractional part of x, and phi=(1+sqrt(5))/2. - Hans J. H. Tuenter, Jul 20 2025

a(n) = A072354(n+1)-A072354(n), a first-order difference. - Hans J. H. Tuenter, Jul 20 2025