A386758 Number of decimal digits in the n-th Lucas number.

1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17

Offset

0,6

Comments

As F(n)<=L(n), the number of decimal digits of the Lucas number L(n) is at least as large as the number of decimal digits of the Fibonacci number F(n). Furthermore, the difference is at most one. The indices for which the difference is one is A386760.

Links

Hans J. H. Tuenter, Table of n, a(n) for n = 0..10000

Formula

a(n) = A055642(A000032(n)).

a(n) = 1 + floor(n*log_10(phi)), where log_10(phi) = A097348, and phi = (1+sqrt(5))/2 = A001622.

Example

L(0)=2 has one digit, so that a(0)=1; L(5)=11 has two digits, so that a(5)=2.

Maple

a:= n-> 1+floor(n*log[10]((1+sqrt(5))/2)):
seq(a(n), n=0..81);

Mathematica

a[n_] := IntegerLength[LucasL[n]]; Array[a, 100, 0] (* Amiram Eldar, Aug 16 2025 *)

Crossrefs

Cf. A000032, A001622, A055642, A060384, A097348.

Number of digits of L(p^n): A094057 (p=2), A114469 (p=10).

Sequence in context: A002266 A075249 A008648 * A154099 A105511 A187183

Adjacent sequences: A386755 A386756 A386757 * A386759 A386760 A386761

Keyword

base,nonn,easy

Author

Hans J. H. Tuenter, Aug 06 2025

Status

approved