I have been looking at Bernoulli's triangle and it's relation to the OEIS sequences that arrive. This picture of Bernoulli's triangle shows the sequences on the Bernoulli's triangle that are the results of various, well known problems throughout math. It's particularly interesting that they can be derived with a very simple pattern across the Bernoulli triangle. However, after the Moser Circle problem, the sequences no longer solve interesting problems and instead are just the continuation of this pattern (a sequence of sequences if you will).
In math, my understanding is that you explore a problem and can sometimes discover a solution that relates to other areas giving key insights to different fields. However, I was wondering if there are techniques or uses for doing the reverse?
How would you go about fabricating an interesting problem, where the result is the next sequence: A006261
Has this kind of problem solving been useful in the past? I'd be interested in related mathematical history or techniques that have been used.