On functions expressible as words on a pair of Beatty sequences

Ballot, Christian

Journal of integer sequences, Tome 20 (2017) no. 4.

Résumé

Summary: Let $a(n)=\lfloor n{\alpha}\rfloor$ and $b(n)=\lfloor n{\alpha}^2\rfloor$, where ${\alpha}=\frac{1+\sqrt{5}}2$. Then a theorem of Carlitz et al. states that each function $f$, composed of several $a$'s and $b$'s, can be expressed in the form $c_{1}a + c_{2}b - c_{3}$, where $c_{1}$ and $c_{2}$ are consecutive Fibonacci numbers determined by the numbers of $a$'s and of $b$'s composing $f$ and $c_{3}$ is a nonnegative constant. We provide generalizations of this theorem to two infinite families of complementary pairs of Beatty sequences. The particular case involving `Narayana' numbers is examined in depth. The details reveal that $x_n= \lfloor{\alpha}^3\lfloor{\alpha}^3\lfloor\cdots\lfloor{\alpha}^3\rfloor\cdots\rfloor\rfloor\rfloor$, with $n$ nested pairs of $\lfloor\;\rfloor$, is a 7th-order linear recurrence, where ${\alpha}$ is the dominant zero of $x^{3} - x^{2} - 1$.

Détail

Classification : 11B83, 11B37, 11B39
Keywords: Beatty sequence, Wythoff pair, integer part, linear recurrence