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.