Equivalence of Definitions of Upper Wythoff Sequence

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Theorem

The following definitions of the upper Wythoff sequence are equivalent:

Definition 1

The upper Wythoff sequence is the complementary Beatty sequence on the golden section $\phi$.

Definition 2

The upper Wythoff sequence is the Beatty sequence on the square $\phi^2$ of the golden section $\phi$.


Proof

From Beatty's Theorem, the Beatty sequences $\mathcal B_r$ and $\mathcal B_s$ are complementary if and only if:

$\dfrac 1 r + \dfrac 1 s = 1$

It remains to be demonstrated that this holds for $r = \phi$ and $s = \phi^2$.


Thus:

\(\ds \dfrac 1 \phi + \dfrac 1 {\phi^2}\) \(=\) \(\ds \dfrac {\phi + 1} {\phi^2}\)
\(\ds \) \(=\) \(\ds \dfrac {\phi^2} {\phi^2}\) Square of Golden Mean equals One plus Golden Mean
\(\ds \) \(=\) \(\ds 1\)

Hence the result.

$\blacksquare$