Definition:Wythoff Sequence
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Definition
Lower Wythoff Sequence
The lower Wythoff sequence is the Beatty sequence on the golden section $\phi$.
It starts:
- $0, 1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, \ldots$
Upper Wythoff Sequence
The upper Wythoff sequence is the Beatty sequence on the square $\phi^2$ of the golden section $\phi$.
The upper Wythoff sequence starts:
- $0, 2, 5, 7, 10, 13, 15, 18, 20, 23, 26, 28, 31, 34, \ldots$
Also see
Source of Name
This entry was named for Willem Abraham Wythoff.
Historical Note
The upper Wythoff sequence and lower Wythoff sequence were introduced in $1926$ by Samuel Beatty in a much-cited puzzle page: Problems for Solutions: 3173-3180 (Amer. Math. Monthly Vol. 33: p. 159) www.jstor.org/stable/2300153.
Their names originate from the fact that, in the form of Wythoff pairs, they form the winning combinations of Wythoff's game.
Sources
- 1926: Samuel Beatty: Problems for Solutions: 3173-3180 (Amer. Math. Monthly Vol. 33: p. 159) www.jstor.org/stable/2300153
- 1927: Samuel Beatty: Solutions: 3177 (Amer. Math. Monthly Vol. 34: p. 159) www.jstor.org/stable/2298716