

A097348


Decimal expansion of arccsch(2)/log(10).


2



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
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OFFSET

1,1


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 int(log10(F(10^n))) = int(1/2*log10(5) + 10^n*log10(phi)). Similarly L(n) tends to phi^n, so the number of digits of L(10^n) is given by int(10^n*log10(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


LINKS

Table of n, a(n) for n=1..104.
Eric Weisstein's World of Mathematics, Fibonacci Number
Eric Weisstein's World of Mathematics, Lucas Number


EXAMPLE

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 *)


CROSSREFS

Cf. A000045, A068070.
Sequence in context: A287542 A288197 A181592 * A271034 A185965 A106193
Adjacent sequences: A097345 A097346 A097347 * A097349 A097350 A097351


KEYWORD

easy,nonn,cons


AUTHOR

Ed Pegg Jr, Aug 06 2004


STATUS

approved



