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I remember I have seen a classical example of a mistake, where someone was convinced that a sequence defined somehow had a close form, which did in turn work until some very high $n$. I think the first $n$ to fail was 3 or 5 digits.

Is anyone aware of what is on my mind?

Coolwater
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There's a huge list here of apparently true things that eventually fail. Maybe you can find what you were looking for here?