Previously on this subreddit, I posted about an idea I had for a combinatoric integer sequence motivated by the brace notation for sets.

I ended up resorting to AI as a tool for exploring this, taking every precaution to make sure it doesn't hallucinate.

I've written everything up in a draft report, here https://app.box.com/s/n6txr68t4t4mwqzl2arimsiuja0977ag

The report begins with a more formal mathematical statement of the question, followed by why I think it's mathematically interesting. I used AI to research applicable terminology, but as the human I always made the final decision about how to express an idea, and never just copied words without understanding them.

It then contains numerical results. I never got numbers directly from the AI. Instead, I got the AI to generate Python code, and from the Python code I got the numbers. This prevents hallucinations.

I also asked the AI to explain the mathematics behind its results, in a beginner friendly manner. I actually learned a lot from this, and my notes from its explanation are included in an appendix (Appendix II). It did hallucinate a fair bit along the way, but I was there to interrogate its responses and make sure it made sense. (This actually led to some lost sleep.) This appendix is the most mathematically meaty part of my report.

The numerical results section actually contains several results that might be of interest to the OEIS, and to members of the seqfan mailing list. Neither of which I'm signed up with, and for several reasons I don't really want to, but if anyone currently active here is a member, let's talk.

It's late in my timezone so I'll be off to bed, and will check on things in the morning. I really hope all of this is interesting to someone. It seems pretty cool to me.

Each number on the left follows the same hidden rule to produce the number on the right. Solve for the missing value from the pattern shown by the other examples. Game: Rule Hunter — https://play.google.com/store/apps/details?id=com.makenumber

Each cluster of three numbers follows the same hidden rule to produce the middle number. Solve for the missing value from the pattern shown by the other cluster. Game: Rule Hunter — https://play.google.com/store/apps/details?id=com.makenumber

Each pair of numbers produces the number below it by the same hidden rule. Solve for the missing number from the pattern shown by the other pairs. Game: Rule Hunter — https://play.google.com/store/apps/details?id=com.makenumber

Each diamond follows the same simple rule. What is the missing number? From my free Android game Rule Hunter: https://play.google.com/store/apps/details?id=com.makenumber