# 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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