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army1987 comments on Beautiful Math - Less Wrong
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This is obvious after you learn calculus. The "nth difference" corresponds to nth derivative (a sequence just looks at integer points of a real-valued function), so clearly a polynomial of degree n has constant nth derivative. It would be even more accurate to say that an nth antiderivative of a constant is precisely a degree n polynomial.
Iterated finite differences correspond to derivatives in some non-obvious way I can't remember (and can't be bothered to find out).