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 A097348 #49 Aug 22 2025 15:51:00 %S A097348 2,0,8,9,8,7,6,4,0,2,4,9,9,7,8,7,3,3,7,6,9,2,7,2,0,8,9,2,3,7,5,5,5,4, %T A097348 1,6,8,2,2,4,5,9,2,3,9,9,1,8,2,1,0,9,5,3,5,3,9,2,8,7,5,6,1,3,9,7,4,1, %U A097348 0,4,8,5,3,4,9,6,7,4,5,9,6,3,2,7,7,6,5,8,5,5,6,2,3,5,1,0,3,5,3,5,1,4,5,0 %N A097348 Decimal expansion of arccsch(2)/log(10). %C A097348 First n terms give number of digits of Fibonacci(10^n), except that it can be off by 1. This is a highly compressed sequence. As a result, it can be off by one. The uncompressed version goes like this: 2, 21, 209, 2090, 20899, 208988, 2089877, 20898764, 208987640, 2089876403, ... (see A068070). Fibonacci(10) = 55 has 2 digits, Fibonacci(100) = 354224848179261915075 has 21 digits and so on. %C A097348 Considering the very good approximation F(n) = 5^(-1/2)*phi^n, the number of digits of F(10^n) is given by floor(log_10(F(10^n))) = floor(-(1/2)*log_10(5) + 10^n*log_10(phi)). Similarly L(n) tends to phi^n, so the number of digits of L(10^n) is given by floor(10^n*log_10(phi)). Both numbers can differ at most by 1. F(n) and L(n) denote the Fibonacci and Lucas numbers, resp. - Christoph Pacher (christoph.pacher(AT)arcs.ac.at), Nov 22 2006 %H A097348 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a> %H A097348 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a> %F A097348 Equals log_10(phi) where phi = (1+sqrt(5))/2. - _Jaroslav Krizek_, Dec 23 2013 %e A097348 0.20898764024997873376... %e A097348 Fibonacci(10^9) has 208987640 decimal digits; %e A097348 Fibonacci(10^21) has 208987640249978733769 decimal digits; %e A097348 Fibonacci(10^27) has 208987640249978733769272089 decimal digits. %p A097348 phi := (1+sqrt(5))/2 ; evalf( log(phi)/log(10)) ; # _R. J. Mathar_, Oct 17 2012 %t A097348 FibonacciDigits[n_] := Ceiling[(2*n*ArcCsch[2] - Log[5])/Log[100]] %t A097348 RealDigits[ArcCsch[2]/Log[10], 10, 105][[1]] (* _Vaclav Kotesovec_, Aug 09 2015 *) %o A097348 (PARI) solve(x=.1,1,sinh(x)-.5)/log(10) \\ _Charles R Greathouse IV_, Aug 04 2020 %o A097348 (PARI) log((1+sqrt(5))/2)/log(10) \\ _Charles R Greathouse IV_, Aug 04 2020 %Y A097348 Cf. A000045, A001622, A068070, A114469. %K A097348 easy,nonn,cons,changed %O A097348 0,1 %A A097348 _Ed Pegg Jr_, Aug 06 2004 %E A097348 Offset corrected by _Lee A. Newberg_, Oct 13 2022