A097348 - OEIS (original) (raw)

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, 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, 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

COMMENTS

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.

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

Decimal expansion of log_10(phi) = log(phi) / log(10), where phi = golden ratio = (1 + sqrt(5))/2 = A001622. - Jaroslav Krizek, Dec 23 2013

EXAMPLE

0.20898764024997873376...

Fibonacci(10^9) has 208987640 decimal digits;

Fibonacci(10^21) has 208987640249978733769 decimal digits;

Fibonacci(10^27) has 208987640249978733769272089 decimal digits.

MAPLE

phi := (1+sqrt(5))/2 ; evalf( log(phi)/log(10)) ; # R. J. Mathar, Oct 17 2012

MATHEMATICA

FibonacciDigits[n_] := Ceiling[(2*n*ArcCsch[2] - Log[5])/Log[100]]

RealDigits[ArcCsch[2]/Log[10], 10, 105][[1]] (* Vaclav Kotesovec, Aug 09 2015 *)