It is well known that every positive integer can be represented uniquely as a sum of distinct, nonconsecutive Fibonacci numbers (see, e.g., Brown . This representation is called the Zeckendorf representation of the positive integer. Other Zeckendorf-type representations where the Fibonacci numbers are not necessarily consecutive are possible. Brown  considers one where a maximal number of distinct Fibonacci numbers are used rather than a minimal number.
Bunder, M. W. (1992). Zeckendorf representations using negative fibonacci numbers. The Fibonacci Quarterly: a journal devoted to the study of integers with special properties, 30 (2), 111-115.