A nice functional equation

Algebra Level 3

A function is defined for all values of \({x}\) such that: \({f}\)(\({x}\)) = \({f}\)(\({x+1}\)) + \({f}\)(\({x-1}\)) and the first two values, \({f}\)(\({1}\)) and \({f}\)(\({2}\)) are arbitrarily chosen (they can take any value). What is the sum of all the possible fundamental periods of the function? [i.e. 1,2,3,1,2,3.... has a cycle of 3) ? (For example, if the function can have cycles of period 1,2 and 3, then the sum would be 6)

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