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Number of digits in 10^n-th Fibonacci number.
3

%I #16 Feb 28 2023 04:08:52

%S 1,2,21,209,2090,20899,208988,2089877,20898764,208987640,2089876403,

%T 20898764025,208987640250,2089876402500,20898764024998,

%U 208987640249979,2089876402499787,20898764024997874,208987640249978734

%N Number of digits in 10^n-th Fibonacci number.

%C a(n)/10^n tends towards log_10((1+sqrt(5))/2) = 0.2089876402499787337692... - _Vladeta Jovovic_, Mar 09 2002

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>.

%F 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

%t Table[ Floor[ Log[10, Fibonacci[10^n]] + 1], {n, 0, 7} ]

%t 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 FibonacciDigits[n_] := Ceiling[(2*n*ArcCsch[2] - Log[5])/Log[100]]; Table[ FibonacciDigits[10^n], {n, 19}]

%Y Cf. A097348.

%K nonn,base

%O 0,2

%A _Robert G. Wilson v_, Feb 26 2002

%E Corrected by _Vladeta Jovovic_, Sep 01 2004