Zeckendorf Representation of Integer shifted Right

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

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


$\floor {\, \cdot \,}$ 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 $k_1, k_2, \ldots, k_r$.


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


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


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.