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On Functions Expressible as Words
on a Pair of Beatty Sequences Christian Ballot
Département de Mathématiques et Informatique
Université de Caen-Normandie
France
mailto:christian.ballot@unicaen.frchristian.ballot@unicaen.fr

in

Abstract:

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 Kimberling states that each function f, composed of several a's and b's, can be expressed in the form c₁a+c₂b-c₃, where c₁ and c₂ are consecutive Fibonacci numbers determined by the numbers of a's and of b's composing f and c₃ 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 x³-x²-1.