A114469 Number of decimal digits in Lucas(10^n).

1, 3, 21, 209, 2090, 20899, 208988, 2089877, 20898765, 208987641, 2089876403, 20898764025, 208987640250, 2089876402500, 20898764024998, 208987640249979, 2089876402499788, 20898764024997874, 208987640249978734, 2089876402499787338, 20898764024997873377, 208987640249978733770, 2089876402499787337693, 20898764024997873376928, 208987640249978733769273, 2089876402499787337692721

Offset: 0

Author

Eric W. Weisstein, Nov 30 2005

Keywords

• nonn
• base

Comments

From Hans J. H. Tuenter, Jul 15 2025: (Start)

This sequence can be constructed by taking the first n digits of the decimal expansion of log_10(phi) = A097348 and adding 1. For example,

a(0) = 1,

a(1) = 2+1 = 3,

a(2) = 20+1 = 21,

a(3) = 208+1 = 209,

a(4) = 2089+1 = 2090,

a(5) = 20898+1 = 20899.

Alternatively, a(n)-1 is the first n digits of A097348. (End)

The individual digits of log_10(phi), and thus sequence A097348, can be extracted using d(n) = a(n)-10*a(n-1)+9. - Hans J. H. Tuenter, Jul 25 2025

Links

• Hans J. H. Tuenter, Table of n, a(n) for n = 0..100
• Eric Weisstein's World of Mathematics, Lucas Number.

Crossrefs

Cf. A068070. - R. J. Mathar, Dec 13 2008

Cf. A000032, A001622, A055642, A097348.

Programs

Mathematica

Table[IntegerLength[LucasL[10^n]], {n, 0, 7}] (* Jean-François Alcover, Jan 24 2018 *)

Formula

Limit_{n->oo} a(n)/10^n = A097348. - Amiram Eldar, Apr 14 2022

a(n) = 1+floor(10^n*log_10(phi)), where phi = (1+sqrt(5))/2, the golden ratio. - Hans J. H. Tuenter, Jul 13 2025.

From Hans J. H. Tuenter, Jul 25 2025: (Start)

a(n) = 1 + Sum_{i=0..n} d(i)*10^(n-i),

a(n) = 10*a(n-1)-9+d(n),

where d(i) = A097348(i-1) is the coefficient of 10^(-i) in the decimal expansion of log_10(phi). (End)

Extensions

a(8)-a(10) from Amiram Eldar, Apr 14 2022

a(11)-a(25) from Hans J. H. Tuenter, Jul 15 2025