A097348 Decimal expansion of arccsch(2)/log(10).

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

Offset: 0

Author

Ed Pegg Jr, Aug 06 2004

Keywords

• easy
• nonn
• cons
• changed

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

Examples

			0.20898764024997873376...
Fibonacci(10^9)  has 208987640 decimal digits;
Fibonacci(10^21) has 208987640249978733769 decimal digits;
Fibonacci(10^27) has 208987640249978733769272089 decimal digits.
		

Links

• Eric Weisstein's World of Mathematics, Fibonacci Number
• Eric Weisstein's World of Mathematics, Lucas Number

Crossrefs

Cf. A000045, A001622, A068070, A114469.

Programs

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 *)

PARI

solve(x=.1,1,sinh(x)-.5)/log(10) \\ Charles R Greathouse IV, Aug 04 2020

PARI

log((1+sqrt(5))/2)/log(10) \\ Charles R Greathouse IV, Aug 04 2020

Formula

Equals log_10(phi) where phi = (1+sqrt(5))/2. - Jaroslav Krizek, Dec 23 2013

Extensions

Offset corrected by Lee A. Newberg, Oct 13 2022