is every function defined on natural numbers considered a sequence? I wonder if every function that operates over the set of natural numbers (or its finite subset ranging from $1$ to $k$) is considered a sequence?

It seems to me that the answer is yes, at least when one looks at the common definition of a sequence:

$$a: \mathbb{J} \to X$$
where: $\mathbb{J}$ is either $\mathbb{N}$ or $\{1, 2,..., k\}$
"We define and as "sequence" or "infinite sequence" any function :f : $ \mathbb N \rightarrow A, A \subseteq \mathbb R$. We also define as "finite sequence"  any function : $f : (1,2,..,n)\rightarrow A, A \subseteq \mathbb R $." Note that $(1,2,...,n)$ is a set $a \subseteq \mathbb N$.
From WolframMathWorld on Sequence:


  A sequence is an ordered set of mathematical objects.


From WolframMathWorld on ordered set:


  [An ordered set is a]n ambiguous term which is sometimes used to mean a partially ordered set and sometimes to mean a totally ordered set.


And I will not quote WolframMathWorld on totally ordered set.



However, there is a site of integer sequence, called the OEIS (On-Line Encyclopedia of Integer Sequences), which have sequences like this:

A000005 (the number of divisors of $n$):


  1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 6, 2, 4, ...




TL;DR:


From WolframMathWorld: the answer is no. A sequence must be increasing.
However, the OEIS (On-Line Encyclopedia of Integer Sequences) also record non-increasing sequences.