This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A068070 #51 Jul 30 2025 12:26:51
%S A068070 1,2,21,209,2090,20899,208988,2089877,20898764,208987640,2089876403,
%T A068070 20898764025,208987640250,2089876402500,20898764024998,
%U A068070 208987640249979,2089876402499787,20898764024997874,208987640249978734,2089876402499787338,20898764024997873377,208987640249978733769
%N A068070 Number of digits in 10^n-th Fibonacci number.
%C A068070 a(n)/10^n tends towards log_10((1+sqrt(5))/2) = 0.2089876402499787337692... = A097348. - _Vladeta Jovovic_, Mar 09 2002
%C A068070 This sequence is very close to A114469, the number of decimal digits in the Lucas number L(10^n). In fact, a(n) = A114469(n) or a(n) = A114469(n) - 1. The first difference occurs at a(1) as F(10) = 55 and L(10) = 123. The next differences occur at indices 8, 9, 16, 21, 23, 24, 27, 34, ... . - _Hans J. H. Tuenter_, Jul 21 2025:
%H A068070 Alois P. Heinz, <a href="/A068070/b068070.txt">Table of n, a(n) for n = 0..1000</a> (first 101 terms from Hans J. H. Tuenter)
%H A068070 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>.
%H A068070 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>.
%F A068070 Can be calculated easily using the fact that Fibonacci(n) is very close to GoldenRatio^n/(2*GoldenRatio-1) (cf. A000045). - _Michael Taktikos_, Aug 11 2004.
%F A068070 a(n) = 1+floor(10^n*log_10(phi)-log_10(5)/2), for n>0, where phi=(1+sqrt(5))/2, the golden ratio. - _Hans J. H. Tuenter_, Jul 08 2025.
%F A068070 a(n) = A055642(A250489(n)). - _Alois P. Heinz_, Jul 09 2025
%p A068070 a:= n-> `if`(n=0, 1, floor(10^n*log[10]((1+sqrt(5))/2)-log[10](5)/2)+1):
%p A068070 seq(a(n), n=0..21); # _Alois P. Heinz_, Jul 30 2025
%t A068070 Table[ Floor[ Log[10, Fibonacci[10^n]] + 1], {n, 0, 7} ]
%t A068070 fib10[n_] := Block[{a = N[Log[10, Sqrt[5]/5], 64], b = N[Log[10, (1 + Sqrt[5])/2], 64]}, Floor[a + 10^n*b] + 1]; Table[ fib10[n], {n, 19}] (* _Robert G. Wilson v_, May 20 2005 *)
%t A068070 FibonacciDigits[n_] := Ceiling[(2*n*ArcCsch[2] - Log[5])/Log[100]]; Table[ FibonacciDigits[10^n], {n, 19}]
%Y A068070 Cf. A000045, A000032, A001622, A055642, A097348, A114469, A250489.
%K A068070 nonn,base
%O A068070 0,2
%A A068070 _Robert G. Wilson v_, Feb 26 2002
%E A068070 Corrected by _Vladeta Jovovic_, Sep 01 2004
%E A068070 a(19)-a(21) from _Hans J. H. Tuenter_, Jul 21 2025