Definition:Wythoff Sequence

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

• Definition:Wythoff Pair

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