Q: What are some examples where a sequence follows a regular pattern out to very large numbers, but not forever? This question is inspired by this old paper by Euler: http://eulerarchive.maa.org/docs/translations/E175en.pdf.
In it, Euler considers a particular sequence, and finds that the next number in the sequence can be computed by a certain recurrence relation. He doesn't have a proof, but verifies it manually up to a a large index, and is convinced of its veracity.
Much later, it was proven that Euler's conjecture was true.
This got me thinking, have there been cases where a sequence was thought to follow some pattern for a long time, but it was eventually shown that the sequence deviates from said pattern? I am not talking about questions that are still open, only those conclusively shown to be false.
