Let
![$a(n)=\lfloor n{\alpha}\rfloor$](abs/img1.gif)
and
![$b(n)=\lfloor n{\alpha}^2\rfloor$](abs/img2.gif)
,
where
![${\alpha}=\frac{1+\sqrt{5}}2$](abs/img3.gif)
.
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
![$x_n=
\lfloor{\alpha}^3\lfloor{\alpha}^3\lfloor\cdots\lfloor{\alpha}^3\rfloor\cdots\rfloor\rfloor\rfloor$](abs/img4.gif)
,
with
n nested
pairs of
![$\lfloor\;\rfloor$](abs/img5.gif)
,
is a 7th-order linear recurrence, where
![${\alpha}$](abs/img6.gif)
is the
dominant zero of
x3 -
x2 - 1.
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.