A060384 Number of decimal digits in n-th Fibonacci number.

1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18

Offset: 0

Author

Labos Elemer, Apr 03 2001

Keywords

• base
• nonn

Links

• Harry J. Smith, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000045, A022307, A001605, A060319, A060320, A051694, A050815.

Cf. A261585.

Programs

Haskell

a060384 = a055642 . a000045  -- Reinhard Zumkeller, Mar 09 2013

Maple

with(combinat): a:=n->nops(convert(fibonacci(n),base,10)): 1,seq(a(n),n=1..100); # Emeric Deutsch, May 19 2007

Mathematica

Table[IntegerLength@ Fibonacci@ n, {n, 0, 84}] /. 0 -> 1 (* or *)
Table[Floor[n Log10@ GoldenRatio - Log10@ 5/2] + 1, {n, 0, 84}] /. 0 -> 1 (* Michael De Vlieger, Jul 04 2016 *)

PARI

print1("1, 1, "); gold=(1+sqrt(5))/2; for(n=2,100,print1(floor((n*log(gold)-log(5)/2)/log(10))+1", ")) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

PARI

a(n) = #Str(fibonacci(n)); \\ Michel Marcus, Jul 04 2016

Formula

a(n) = floor(n*log(tau)/log(10)) +0 or +1 where tau is the golden ratio. - Benoit Cloitre, Oct 29 2002. [Corrected by Hans J. H. Tuenter, Jul 07 2025].

a(n) = floor(n*log_10(gold) - log_10(5)/2) + 1 for n >= 2, where gold is (1+sqrt(5))/2. - Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007

a(n) = A055642(A000045(n)). - Reinhard Zumkeller, Mar 09 2013

Extensions

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), May 01 2007