Journal of Integer Sequences, Vol. 20 (2017), Article 17.4.2

On Functions Expressible as Words on a Pair of Beatty Sequences

Christian Ballot
Département de Mathématiques et Informatique
Université de Caen-Normandie


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 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.

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(Concerned with sequences A017898 A078012.)

Received July 26 2016; revised versions received January 17 2017; January 28 2017. Published in Journal of Integer Sequences, January 28 2017. Minor revision, July 30 2017.

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