Beatty sequence

The integer sequence

{\\cal B}(\\alpha,\\alpha^{\\prime}):=\\left(\\left\\lfloor\\frac{n-\\alpha^{\\prime}}{%\\alpha}\\right\\rfloor\\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(\\left\\lceil\\frac{n-\\alpha^{%\\prime}}{\\alpha}\\right\\rceil\\right)_{n=1}^{\\infty}

is called a ceiling Beatty sequence.

References

[1: ] M. Lothaire, , 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, VeroniqueBruyere, Christiane Frougny, Filippo Mignosi, Antonio Restivo,Christophe Reutenauer, Dominique Foata, Guo-Niu Han, JacquesDesarmenien, Volker Diekert, Tero Harju, Juhani Karhumaki andWojciech Plandowski;With a preface by Berstel and Perrin. http://www.ams.org/mathscinet-getitem?mr=1905123MR 1905123