cp's OEIS Frontend

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

%I A105565 #22 Aug 05 2025 18:12:09
%S A105565 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,
%T A105565 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,
%U A105565 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
%N A105565 a(n) = if (exactly 5 Fibonacci numbers exist with exactly n digits) then 1, otherwise 0.
%C A105565 The sequence is almost periodic, see also A105566;
%C A105565 a(n) = 1 - A105563(n), for n>1.
%C A105565 a(n) = A050815(n) - 4, for n>1. - _Hans J. H. Tuenter_, Jul 29 2025
%H A105565 Robert Israel, <a href="/A105565/b105565.txt">Table of n, a(n) for n = 1..10000</a>
%H A105565 Igor Szczyrba, Rafał Szczyrba, and Martin Burtscher, <a href="https://www.emis.de/journals/JIS/VOL18/Szczyrba/sz3.html">Analytic Representations of the n-anacci Constants and Generalizations Thereof</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.4.5.
%H A105565 Jürgen Spilker, <a href="https://www.researchgate.net/publication/251340897_Die_Ziffern_der_Fibonacci-Zahlen">Die Ziffern der Fibonacci-Zahlen</a>, Elemente der Mathematik 58 (Birkhäuser 2003).
%H A105565 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%H A105565 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPeriodicFunction.html">Almost Periodic Function</a>
%F A105565 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
%e A105565 The Fibonacci numbers with two decimal digits are 13, 21, 34, 55, 89; a total of five, so that a(2)=1.
%p A105565 n:= 1: count:= 2: a:= 0: b:= 1:
%p A105565 for m from 2 while n < 101 do
%p A105565   c:= b; b:= a+b; a:= c;
%p A105565   s:= ilog10(b)+1;
%p A105565   if s = n then count:= count+1
%p A105565   else
%p A105565     if count = 5 then A[n]:= 1 else A[n]:= 0 fi;
%p A105565     count:= 1; n:= s
%p A105565   fi
%p A105565 od:
%p A105565 seq(A[i],i=1..100); # _Robert Israel_, Dec 17 2018
%Y A105565 Cf. A050815, A060384, A000045.
%K A105565 nonn,base
%O A105565 1,1
%A A105565 _Reinhard Zumkeller_, Apr 14 2005