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 $\BB_r$ and $\BB_s$ 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.
$\blacksquare$