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3/ I used to point this sort of thing out all the time to my 2nd grade math teacher (when we did these sequences a lot), and became obsessed w generating multiple sequences that could give rise to the same sample but diverged laterhttps://twitter.com/josecamoessilva/status/1482050297220329476 …
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Any number X of circles is correct; clearly the sequence enumerates the zeros of the following polynomial* in ascending order: (x-1)(x-1)(x-2)(x-X). *As valid as a sequence as the Fibonacci sequence. Unless the game is "guess what the test grader wants."
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Any number? You’d have a hard time convincing me of any other than the 35,045 sequences starting 1,1,2 that have show up in the literature before: https://oeis.org/search?q=1%2C1%2C2&language=english&go=Search …
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it could also be a factorial sequence (0!, 1!, 2!...)
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this represents a more subtle, general point too, which is that while pattern recognition can be a helpful heuristic in math, it is also very frequently a misleading one b/c at low numbers of instances, many distinct patterns are identical
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