Oddly, I derived that fraction through a means completely unrelated to generating functions, and only realised later on that its denominator had a suspicious resemblance to the Fibonacci difference equation. This kind of interconnectedness is one of the most wonderful things about maths, in my book.

Hm... apparently that number isn't in The On-Line Encyclopedia of Integer Sequences (http://www.research.att.com/~njas/sequences/). Why don't you submit it?

I considered doing so, but thought it would be more proper if your name were attached to it.

Decimal expansions of certain numbers are certainly some of the sequences stored there; as, for example, pi (as sequence A000796) (http://www.research.att.com/projects/OEIS?Anum=A000796) or the decimal expansion of 1/7 (as sequence A02080) (http://www.research.att.com/projects/OEIS?Anum=A02080). So why not 100/9899, especially since you can cross-reference it to other sequences already in the database, specifically the Fibonacci sequence (A000045) (http://www.research.att.com/projects/OEIS?Anum=A000045).

My maths is a little rusty unfortunately, so I'm wondering if it actually keeps generating the fibonnaci sequence endlessly or whether it has some limit (so far, up to 40 dt is seems to work fine) and why that might be the case.

So yeah, if you want to do more posts on funky stuff like that, I'm all for it...