Geodesic
Some Properties of Beatty Sequences I*
Canadian mathematical bulletin, Tome 2 (1959) no. 3, pp. 190-197

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Two sequences of natural numbers are said to be complementary if they contain all the positive integers without repetition or omission. S. Beatty [l] observed that the sequences (1) (2) (where square brackets denote the integral part function) are complementary if and only if α > 0 and α is irrational. We call the pair (1),(2) Beatty sequences of argument α. 
Connell, Ian G. Some Properties of Beatty Sequences I*. Canadian mathematical bulletin, Tome 2 (1959) no. 3, pp. 190-197. doi: 10.4153/CMB-1959-025-0
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[1] 1. Beatty, S., Amer. Math. Monthly, 33 (1926), 159 (problem); solutions, ibid., 34(1927), 159. Google Scholar

[2] 2. Connell, I.G., A generalization of Wythoff's game, Can. Math. Bull. 2 (1959), Google Scholar

[3] 3. Sierpinski, W., Bull. Inter. Acad. Sc., Cracovie, 11(1909), 725-727. Google Scholar

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