Because these have been so successful in the past, I had to bring up my own. In fact, this puzzle was inspired by Find the next number even though I (and I guess everybody but the author) still don't know the intended solution yet. Anyway, here it is:
The British inventor Jackemias Muff invented (and build) a number-evolving machine in 1817. (It was a cold, windy winter's day in Manchester, the Sunday before boxing-day.) You could feed it with any number, and it would produce an unambiguously determined, endless series of follow-up numbers.
For example, if you feed it with "8" you would get
If you feed it with "28" instead, you would get
Can you explain how his machine works and build a copy?
Victory condition: A correct answer is able to reproduce both series starting from the same seed values and can successfully predict the next number in each series.
I do not have proof that Muff's machine is unique, although I believe it is.
I will add hints over time, one is given as a starter:
In 1817, computers have not yet been invented. (Well, 'computer' was still a job-description those days..)