A nice functional equation

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)