Representations found by the greedy algorithm

[Kimberling, Clark]

Suppose x=(1+sqrt(5))/2.  The greedy algorithm finds that every positive
integer N has a representation

N = c0/x + c1/x^2 + c2/x^3 + ...

Can someone prove that c1, c2, c3, ... are all 0 except for finitely
many 1's?

Examples:

4 = 6/x + 1/x^3 + 1/x^6

5 = 8/x + 1/x^6

6 = 9/x + 1/x^2 + 1/x^6

22 = 35/x + 1/x^3 + 1/x^5 + 1/x^7 + 1/x^10.

Clark Kimberling