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

Département de Mathématiques et Informatique

Université de Caen-Normandie

France

**Abstract:**

Let
and
,
where
.
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
,
with *n* nested
pairs of
,
is a 7th-order linear recurrence, where
is the
dominant zero of *x*^{3} - *x*^{2} - 1.

(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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