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I am wondering if there's some tip for guessing the degree of a function or if it really is just a guess (assuming one doesn't know all the inner workings of what produced the number in the first place).

I ask because if I guess the degree and get the recurrence coefficients, sometimes if I undershoot the degree, the coefficients are integers but they turn out to be wrong.

Is it really just guess and check? Guess the degree, get the coefficients, test it against the sequence, if it doesn't work, increase the degree and try again, etc?

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If the function is a polynomial, as I'm assuming, finite differences is what you're looking for.

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The question is unclear. If you are saying that you have the terms of a sequence produced by an unknown (constant-coefficient, linear) recurrence and you want to work out the degree of the recurrence (as opposed to the degree of the solution of a recurrence), there is a theorem of Kronecker that says that $c_0,c_1,\dots$ satisfies such a recurrence if and only if $$\det\pmatrix{c_0&c_1&\dots&c_m\cr c_1&c_2&\dots&c_{m+1}\cr\dots&\dots&\dots&\dots\cr c_m&c_{m+1}&\dots&c_{2m}\cr}$$ has determinant zero for all $m$ sufficiently large. What's more, once you've found the $m$ that works, you can work out the recurrence and its degree. Details are given in Salem, Algebraic Numbers and Fourier Analysis, pages 5 to 7.

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