Formula for cold positions

Statement

Wythoff discovered that the cold positions follow a regular pattern determined by the golden ratio. Specifically, if k is any natural number and

$$n_k = \lfloor k \phi \rfloor = \lfloor m_k \phi \rfloor -m_k \,$$

$$m_k = \lfloor k \phi^2 \rfloor = \lceil n_k \phi \rceil = n_k + k \,$$

where φ is the golden ratio and we are using the floor function, then (n_k, m_k) is the k^th cold position. These two sequences of numbers are recorded in the Online Encyclopedia of Integer Sequences as and , respectively.

The two sequences n_k and m_k are the Beatty sequences associated with the equation

$$\frac{1}{\phi} + \frac{1}{\phi^2} = 1 \,.$$

As is true in general for pairs of Beatty sequences, these two sequences are complementary: each positive integer appears exactly once in either sequence.