Generating Fraction for Lucas Numbers/Corollary

Corollary to Generating Fraction for Lucas Numbers
The fraction:
 * $\dfrac {1999} {998 \, 999}$

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.

Proof
By Generating Function for Lucas Numbers:

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

For $n = 3$: