# Definition:Zeckendorf Representation

(Redirected from Definition:Fibonacci Number System)

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## Definition

**Zeckendorf representation** is a system for representing a positive integer $m$ by a sequence of digits which are the indices of a sequence of $r$ Fibonacci numbers:

- $n := k_1 k_2 k_3 \ldots k_r$

where:

- $n = F_{k_1} + F_{k_2} + F_{k_3} + \cdots + F_{k_r}$
- $k_1 \gg k_2 \gg k_3 \gg \cdots \gg k_r \gg 0$

where $n \gg k$ denotes that $n \ge k + 2$.

## Also known as

Some sources give this as the **Fibonacci number system**.

## Also see

- Results about
**Zeckendorf representation**can be found here.

## Source of Name

This entry was named for Edouard Zeckendorf.

## Sources

- 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers: Exercise $34$