Definition:Zeckendorf Representation

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

• Zeckendorf's Theorem

• 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 $34$