Zeckendorf Representation of Integer shifted Right

Theorem

Let $f: \R \to \R$ be the real function defined as:

$\forall x \in \R: \map f x = \floor {x + \phi^{-1} }$

where:

$\floor {\, \cdot \,}$ denotes the floor function

$\phi$ denotes the golden mean.

Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $n$ be expressed in Zeckendorf representation:

$n = F_{k_1} + F_{k_2} + \cdots + F_{k_r}$

with the appropriate restrictions on $k_1, k_2, \ldots, k_r$.

Then:

$F_{k_1 - 1} + F_{k_2 - 1} + \cdots + F_{k_r - 1} = \map f {\phi^{-1} n}$

Proof

Follows directly from Zeckendorf Representation of Integer shifted Left, substituting $F_{k_j - 1}$ for $F_{k_j}$ throughout.

$\blacksquare$

Historical Note

According to Donald E. Knuth in his The Art of Computer Programming: Volume 1: Fundamental Algorithms, 3rd ed. of $1997$, this result was demonstrated by Yuri Vladimirovich Matiyasevich in $1990$, but no further details are given.

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 $41$