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dlthomas comments on Beautiful Math - Less Wrong
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Back in high school I discovered this by accident (yes, I was really bored!). I suppose it's nothing new, but it turns out that this works for more than simple squares and cubes:
Given any sequence of numbers, keep finding differences of differences until you hit a constant; the number of iterations needed is the maximum exponent in the formula that produced the numbers. That is, this works even if there are other terms, regardless of whether any or all terms have coefficients other than 1.
Your procedure (though not necessarily your result) breaks for e^x
Really for non-polynomials, and I think that was implied by the phrasing.
I agree that it's implied by working out the logic and finding that it doesn't apply elsewhere. I disagree that it is implied by the phrasing.
doesn't seem to restrict it, and though I suppose
implies that there is a "maximum exponent in the formula" and with slightly more reasoning (a number of iterations isn't going to be fractional) that it must be a formula with a whole number maximum exponent, I don't see anything that precludes, for instance, x^2 + x^(1/2), which would also never go constant.
Sorry, I was using the weak "implies", and probably too much charity.
And I usually only look at this sort of thing in the context of algorithm analysis, so I'm used to thinking that x squared is pretty much equal to 5 x squared plus 2 log x plus square root of x plus 37.