# 1/9899

Here’s a nice little exercise:

1. Find the first 20 or so digits of 1/9899.

2. Marvel at the beauty.

3. Explain.

3. Wow wow beautiful exercise!! so.. generating functions allow us to create any (¿? at least coming from a recursive sequence) “special” number we want! Let $a_0$ and $a_1$ two integers and construct a sequence satisfying the recurrence formula $a_{n+1}=a_{n}+a_{n-1}$. This sequence has the generating function $F(x)=\frac{a_0+(a_1-a_0)x}{1-x-x^2}$. Show time! Take $a_0=a_1=1$ and see which numbers are $F(1/10)$, $F(1/100)$… (well up to a decimal scaling!)