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

%I A105563 #20 Sep 03 2025 19:43:15
%S A105563 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,
%T A105563 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,
%U A105563 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
%N A105563 a(n) = if (exactly 4 Fibonacci numbers exist with exactly n digits) then 1, otherwise 0.
%C A105563 The sequence is almost periodic, see also A105564;
%C A105563 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
%H A105563 Hans J. H. Tuenter, <a href="/A105563/b105563.txt">Table of n, a(n) for n = 1..10000</a>
%H A105563 Jürgen Spilker, <a href="http://dx.doi.org/10.5169/seals-8480">Die Ziffern der Fibonacci-Zahlen</a>, Elemente der Mathematik, 58(1):26-33, 2003.
%H A105563 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%H A105563 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPeriodicFunction.html">Almost Periodic Function</a>
%F A105563 a(n) = 1 - A105565(n), for n>1.
%F A105563 a(n) = 5 - A050815(n), for n>1. - _Hans J. H. Tuenter_, Aug 28 2025
%F A105563 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
%t A105563 If[#==4,1,0]&/@Tally[IntegerLength/@Fibonacci[Range[500]]][[;;,2]] (* _Harvey P. Dale_, Nov 15 2023 *)
%Y A105563 Cf. A000045, A050815, A060384, A105565, A001622.
%K A105563 nonn,base,changed
%O A105563 1,1
%A A105563 _Reinhard Zumkeller_, Apr 14 2005