A105563 -id:A105563 - cp's OEIS frontend

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

Patrick De Geest, Oct 15 1999

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

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

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.

See A098842 for another version.

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

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

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

A105565 a(n) = if (exactly 5 Fibonacci numbers exist with exactly n digits) then 1, otherwise 0.

Original entry on oeis.org

0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Apr 14 2005

The sequence is almost periodic, see also A105566;
a(n) = 1 - A105563(n), for n>1.
a(n) = A050815(n) - 4, for n>1. - Hans J. H. Tuenter, Jul 29 2025

			The Fibonacci numbers with two decimal digits are 13, 21, 34, 55, 89; a total of five, so that a(2)=1.

Robert Israel, Table of n, a(n) for n = 1..10000
Igor Szczyrba, Rafał Szczyrba, and Martin Burtscher, Analytic Representations of the n-anacci Constants and Generalizations Thereof, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
Jürgen Spilker, Die Ziffern der Fibonacci-Zahlen, Elemente der Mathematik 58 (Birkhäuser 2003).
Eric Weisstein's World of Mathematics, Fibonacci Number
Eric Weisstein's World of Mathematics, Almost Periodic Function

Cf. A050815, A060384, A000045.

n:= 1: count:= 2: a:= 0: b:= 1:
for m from 2 while n < 101 do
  c:= b; b:= a+b; a:= c;
  s:= ilog10(b)+1;
  if s = n then count:= count+1
  else
    if count = 5 then A[n]:= 1 else A[n]:= 0 fi;
    count:= 1; n:= s
  fi
od:
seq(A[i],i=1..100); # Robert Israel, Dec 17 2018

For n>1, a(n) = [{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 29 2025

A105564 Number of blocks of exactly 4 Fibonacci numbers having equal length <= n.

Original entry on oeis.org

0, 0, 0, 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, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 18, 18

Offset: 1

Reinhard Zumkeller, Apr 14 2005

nonn

Lim_{n->infinity} a(n)/n = 5 - 1/log_10((1+sqrt(5))/2) = 0.215....

a(n) = Sum_{k=1..n} A105563(k); a(n) = n - A105566(n).

Juergen Spilker, Die Ziffern der Fibonacci-Zahlen, Elemente der Mathematik 58 (Birkhäuser 2003).

A144601 Christoffel word of slope 3/11.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0

Offset: 0

N. J. A. Sloane, Jan 13 2009

See A144595 for further details.

Different from A105563.

a(n) = a(n-14). - R. J. Mathar, May 30 2025

G.f.: -x^4*(1+x^5+x^9) / ( (x-1)*(1+x^6+x^5+x^4+x^3+x^2+x)*(1+x)*(1-x+x^2-x^3+x^4-x^5+x^6) ). - R. J. Mathar, May 30 2025

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A050815 Number of positive Fibonacci numbers with n decimal digits.

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A105565 a(n) = if (exactly 5 Fibonacci numbers exist with exactly n digits) then 1, otherwise 0.

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A105564 Number of blocks of exactly 4 Fibonacci numbers having equal length <= n.

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A144601 Christoffel word of slope 3/11.

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