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Suppose that $$f(n) + (n+1)^2 = f(n+1),$$

  1. How could I find the original (or family of) function(s) that satisfies this property?
  2. What is the branch of mathematics that deals with equations like this?
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    You're trying to solve a *functional equation*.2011-08-01
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    Assuming you meant $f(n)$ rather than $f(x)$ in your equation, then this is simply a case of a linear recurrence relation (with constant coefficients!), or definition by recursion. There is a branch of mathematics called recursive function theory but it is almost surely not what you're interested in. I suspect you want to know about solutions to more complicated *functional equations* like $f(f(x)) = x$, but I don't think this is a well-understood topic in mathematics yet.2011-08-01
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    With the edit: you now have a *difference equation*, or as Henry says, a recurrence/recursion relation.2011-08-01
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    1. You're looking for an antidifference of a hypergeometric, so Gosper's algorithm will give you the solution.2011-08-01

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