A068070 Number of digits in 10^n-th Fibonacci number.

1, 2, 21, 209, 2090, 20899, 208988, 2089877, 20898764, 208987640, 2089876403, 20898764025, 208987640250, 2089876402500, 20898764024998, 208987640249979, 2089876402499787, 20898764024997874, 208987640249978734, 2089876402499787338, 20898764024997873377, 208987640249978733769

Offset: 0

Author

Robert G. Wilson v, Feb 26 2002

Keywords

• nonn
• base

Comments

a(n)/10^n tends towards log_10((1+sqrt(5))/2) = 0.2089876402499787337692... = A097348. - Vladeta Jovovic, Mar 09 2002

This sequence is very close to A114469, the number of decimal digits in the Lucas number L(10^n). In fact, a(n) = A114469(n) or a(n) = A114469(n) - 1. The first difference occurs at a(1) as F(10) = 55 and L(10) = 123. The next differences occur at indices 8, 9, 16, 21, 23, 24, 27, 34, ... . - Hans J. H. Tuenter, Jul 21 2025:

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000 (first 101 terms from Hans J. H. Tuenter)
• Eric Weisstein's World of Mathematics, Fibonacci Number.
• Eric Weisstein's World of Mathematics, Lucas Number.

Crossrefs

Cf. A000045, A000032, A001622, A055642, A097348, A114469, A250489.

Programs

Maple

a:= n-> `if`(n=0, 1, floor(10^n*log[10]((1+sqrt(5))/2)-log[10](5)/2)+1):
seq(a(n), n=0..21);  # Alois P. Heinz, Jul 30 2025

Mathematica

Table[ Floor[ Log[10, Fibonacci[10^n]] + 1], {n, 0, 7} ]
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 *)
FibonacciDigits[n_] := Ceiling[(2*n*ArcCsch[2] - Log[5])/Log[100]]; Table[ FibonacciDigits[10^n], {n, 19}]

Formula

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.

a(n) = 1+floor(10^n*log_10(phi)-log_10(5)/2), for n>0, where phi=(1+sqrt(5))/2, the golden ratio. - Hans J. H. Tuenter, Jul 08 2025.

a(n) = A055642(A250489(n)). - Alois P. Heinz, Jul 09 2025

Extensions

Corrected by Vladeta Jovovic, Sep 01 2004

a(19)-a(21) from Hans J. H. Tuenter, Jul 21 2025