<s>
In	O
mathematics	O
and	O
theoretical	O
computer	O
science	O
,	O
a	O
k-regular	B-Application
sequence	I-Application
is	O
a	O
sequence	O
satisfying	O
linear	O
recurrence	O
equations	O
that	O
reflect	O
the	O
base-k	O
representations	O
of	O
the	O
integers	O
.	O
</s>
<s>
The	O
class	O
of	O
k-regular	B-Application
sequences	I-Application
generalizes	O
the	O
class	O
of	O
k-automatic	O
sequences	O
to	O
alphabets	O
of	O
infinite	O
size	O
.	O
</s>
<s>
There	O
exist	O
several	O
characterizations	O
of	O
k-regular	B-Application
sequences	I-Application
,	O
all	O
of	O
which	O
are	O
equivalent	O
.	O
</s>
<s>
The	O
formal	O
series	O
definition	O
of	O
a	O
k-regular	B-Application
sequence	I-Application
leads	O
to	O
an	O
automaton	O
characterization	O
similar	O
to	O
Schützenberger	O
's	O
matrix	O
machine	O
.	O
</s>
<s>
The	O
notion	O
of	O
k-regular	B-Application
sequences	I-Application
was	O
first	O
investigated	O
in	O
a	O
pair	O
of	O
papers	O
by	O
Allouche	O
and	O
Shallit	O
.	O
</s>
<s>
Prior	O
to	O
this	O
,	O
Berstel	O
and	O
Reutenauer	O
studied	O
the	O
theory	O
of	O
rational	O
series	O
,	O
which	O
is	O
closely	O
related	O
to	O
k-regular	B-Application
sequences	I-Application
.	O
</s>
<s>
The	O
sequence	O
of	O
Cantor	O
numbers	O
c(n )	O
(	O
)	O
consists	O
of	O
numbers	O
whose	O
ternary	B-Algorithm
expansions	I-Algorithm
contain	O
no	O
1s	O
.	O
</s>
<s>
of	O
numbers	O
whose	O
ternary	B-Algorithm
expansions	I-Algorithm
contain	O
no	O
2s	O
is	O
also	O
2-regular	O
.	O
</s>
<s>
A	O
somewhat	O
interesting	O
application	O
of	O
the	O
notion	O
of	O
k-regularity	O
to	O
the	O
broader	O
study	O
of	O
algorithms	O
is	O
found	O
in	O
the	O
analysis	O
of	O
the	O
merge	B-Algorithm
sort	I-Algorithm
algorithm	O
.	O
</s>
<s>
As	O
a	O
result	O
,	O
the	O
sequence	O
defined	O
by	O
the	O
recurrence	O
relation	O
for	O
merge	B-Algorithm
sort	I-Algorithm
,	O
T(n )	O
,	O
constitutes	O
a	O
2-regular	O
sequence	O
.	O
</s>
<s>
Allouche	O
and	O
Shallit	O
give	O
a	O
number	O
of	O
additional	O
examples	O
of	O
k-regular	B-Application
sequences	I-Application
in	O
their	O
papers	O
.	O
</s>
<s>
k-regular	B-Application
sequences	I-Application
exhibit	O
a	O
number	O
of	O
interesting	O
properties	O
.	O
</s>
<s>
A	O
k-regular	B-Application
sequence	I-Application
takes	O
on	O
finitely	O
many	O
values	O
if	O
and	O
only	O
if	O
it	O
is	O
k-automatic	O
.	O
</s>
<s>
This	O
is	O
an	O
immediate	O
consequence	O
of	O
the	O
class	O
of	O
k-regular	B-Application
sequences	I-Application
being	O
a	O
generalization	O
of	O
the	O
class	O
of	O
k-automatic	O
sequences	O
.	O
</s>
<s>
The	O
class	O
of	O
k-regular	B-Application
sequences	I-Application
is	O
closed	O
under	O
termwise	O
addition	O
,	O
termwise	O
multiplication	O
,	O
and	O
convolution	B-Language
.	O
</s>
<s>
The	O
class	O
of	O
k-regular	B-Application
sequences	I-Application
is	O
also	O
closed	O
under	O
scaling	O
each	O
term	O
of	O
the	O
sequence	O
by	O
an	O
integer	O
λ	O
.	O
</s>
<s>
The	O
nth	O
term	O
of	O
a	O
k-regular	B-Application
sequence	I-Application
of	O
integers	O
grows	O
at	O
most	O
polynomially	O
in	O
n	O
.	O
</s>
