A105565 -id:A105565 - 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

A105563 a(n) = if (exactly 4 Fibonacci numbers exist with exactly n digits) then 1, otherwise 0.

Original entry on oeis.org

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, 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

Offset: 1

Reinhard Zumkeller, Apr 14 2005

The sequence is almost periodic, see also A105564;

Asymptotically, a fraction of 1-alpha=0.215028... of the terms are 1. For the partial sums S(n) = Sum_{k=1..n} a(k), this implies S(n)~(1-alpha)*n. Conjecture: -beta < S(n)-(1-alpha)*n < 1-beta. The constants alpha and beta are as defined in the formula section. - Hans J. H. Tuenter, Aug 28 2025

Hans J. H. Tuenter, Table of n, a(n) for n = 1..10000
Jürgen Spilker, Die Ziffern der Fibonacci-Zahlen, Elemente der Mathematik, 58(1):26-33, 2003.
Eric Weisstein's World of Mathematics, Fibonacci Number
Eric Weisstein's World of Mathematics, Almost Periodic Function

Cf. A000045, A050815, A060384, A105565, A001622.

If[#==4,1,0]&/@Tally[IntegerLength/@Fibonacci[Range[500]]][[;;,2]] (* Harvey P. Dale, Nov 15 2023 *)

a(n) = 1 - A105565(n), for n>1.
a(n) = 5 - A050815(n), for n>1. - Hans J. H. Tuenter, Aug 28 2025
For n>1, a(n) = [{n*alpha+beta}>alpha], where alpha=log(10)/log(phi)-4, beta=log(5)/(2*log(phi))-1, [] is the Iverson bracket, {x}=x-floor(x), denotes the fractional part of x, and phi = (1+sqrt(5))/2 = A001622. - Hans J. H. Tuenter, Aug 28 2025

A105566 Number of blocks of exactly 5 Fibonacci numbers having equal length <= n.

Original entry on oeis.org

0, 1, 2, 2, 3, 4, 5, 5, 6, 7, 8, 9, 9, 10, 11, 12, 13, 13, 14, 15, 16, 16, 17, 18, 19, 20, 20, 21, 22, 23, 24, 24, 25, 26, 27, 27, 28, 29, 30, 31, 31, 32, 33, 34, 34, 35, 36, 37, 38, 38, 39, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 48, 49, 49, 50, 51, 52, 53, 53, 54, 55, 56, 56, 57

Offset: 1

Reinhard Zumkeller, Apr 14 2005

nonn

a(n) = Sum_{k=1..n} A105565(k); a(n) = n - A105564(n);

lim_{n->inf} a(n)/n = 1/log_10((1+sqrt(5))/2) - 4 = 0.784....

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

Eric Weisstein's World of Mathematics, Fibonacci Number

Cf. A105564, A072353, A050815, A060384, A000045.

cp's OEIS Frontend

A050815 Number of positive Fibonacci numbers with n decimal digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A105563 a(n) = if (exactly 4 Fibonacci numbers exist with exactly n digits) then 1, otherwise 0.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A105566 Number of blocks of exactly 5 Fibonacci numbers having equal length <= n.

Views

Author

Keywords

Comments

References

Links

Crossrefs