998999 and fibonacci numbers

1/998999 = 1.001002003005008013021034055089 144 233 377 610 ... × 10^-6

I came across this on twitter and I thought it was interesting. The number 1/998999 contains instances of the Fibonacci numbers.

This works by considering the generating function for the series which has the following closed form:

If we let k = 1000, we get the series appearing in blocks of threes in the decimal representation. If we want higher precision, we can just let k be some higher value.

To extend this idea, my friend and I tried to find the fraction that generates a similar series: Lucas series. It is basically Fibonacci series with the first two terms being 2 and 1 instead of 0 and 1.

The generating function is

using x = 1000 like before, we get the fraction

1999000/998999 = 2.001003004007011018029047076 123 199 322 ...

We can also use this method to generate other interesting sequences as well.

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