The integer sequence $$ {\cal B}(\alpha,\alpha^\prime) := \left( \floor{\frac{n-\alpha^\prime}{\alpha}} \right)_{n=1}^\infty $$ is called the Beatty sequence with density $\alpha$ , slope $\frac{1}{\alpha}$ , offset $\alpha^\prime$ , and $y$ -intercept $\frac{-\alpha^\prime}{\alpha}$ .

Sometimes a sequence of the above type is called a floor Beatty sequence, and denoted ${\cal B}^{(f)}(\alpha,\alpha^\prime)$ , while an integer sequence $$ {\cal B}^{(c)}(\alpha,\alpha^\prime) := \left( \ceiling{\frac{n-\alpha^\prime}{\alpha}} \right)_{n=1}^\infty $$ is called a ceiling Beatty sequence.

References

[1]

M. Lothaire, Algebraic combinatorics on words, vol. 90, Cambridge University Press, Cambridge, 2002, ISBN 0-521-81220-8, available online at http://www-igm.univ-mlv.fr/~berstel/Lothaire. A collective work by Jean Berstel, Dominique Perrin, Patrice Seebold, Julien Cassaigne, Aldo De Luca, Steffano Varricchio, Alain Lascoux, Bernard Leclerc, Jean-Yves Thibon, Veronique Bruyere, Christiane Frougny, Filippo Mignosi, Antonio Restivo, Christophe Reutenauer, Dominique Foata, Guo-Niu Han, Jacques Desarmenien, Volker Diekert, Tero Harju, Juhani Karhumaki and Wojciech Plandowski; With a preface by Berstel and Perrin. MR 1905123