Generating Fraction for Lucas Numbers/Corollary

Corollary to Generating Fraction for Lucas Numbers

$\dfrac {1999} {998 \, 999}$

has a decimal expansion which contains within it the start of the Lucas sequence:

$0 \cdotp 00200 \, 10030 \, 04007 \, 011 \ldots$

and in general, the fraction:

$\dfrac {2 \times 10^n - 1} {10^{2 n} - 10^n - 1}$

contains the Lucas sequence spread out with $n$ digits between each term.

$\ds \sum_{k \mathop \ge 0} L_k 10^{-n k - n}$

$=$

$\ds \frac {2 - 10^{-n} } {1 - 10^{-n} - 10^{-2 n} } \times 10^{-n}$

$\ds $

$=$

$\ds \frac {2 \times 10^n - 1} {10^{2 n} - 10^n - 1}$

The first few terms are contained in the decimal expansion, as long as $L_{k + 1} < 10^n$, where there is no carry.

For $n = 3$:

$\ds \frac {2 \times 10^n - 1} {10^{2 n} - 10^n - 1}$

$=$

$\ds \frac {2000 - 1} {1000000 - 1000 - 1}$

$\ds $

$=$

$\ds \frac {1999} {998999}$

$\blacksquare$