Is sequence related to ordered pairs? I was exploring the meaning of sequence, and that what I found

Let $$ be a non-empty set.
A sequence of elements of $$ is map $$ from $ℕ$ to $$.
That is, the domain of $$ is the whole set $\mathbb{N}$
$$  : ℕ \to  $$
where $_:=()∈$, for all $∈ℕ$.
Recall that $ℕ={1,2,3,…}$

So, my question is, can elements of sequences be treated like pairs? where the first element of the pair is the order and the second element is the value.
 A: The answer is yes if you consider the simple example of a function that generates Pell numbers by iteration. Pairs of these values are needed for Euclid's formula
$$ \qquad A=m^2-k^2\qquad B=2mk \qquad C=m^2+k^2\qquad$$
to generate
Pythagorean triples
where $\quad A-B=\pm1.
\space$
\begin{equation}
\text{The formula}\quad 
\quad m=k+\sqrt{2k^2+(-1)^k}
\quad\text{generates}\quad 1,2,5,12,29,70,\cdots
\end{equation}
Each $\space n^{th}\space\space  k$-value generates an $m$-value and each $m$-value becomes the $\space (n+1)^{th} \space\space k$-value.
\begin{align*}
k=1\qquad &\implies m=(1+\sqrt{2(1)^2+(-1)^1}\space)\big)=2\space  & F(2,1)=(3,4,5)\\
k=2\qquad  &\implies m=(2+\sqrt{2(2)^2+(-1)^2}\space)\big)=5\space  & F(5,2)=(21,20,29)\\
k=5\qquad  &\implies m=(5+\sqrt{2(5)^2+(-1)^5}\space)\big)=12\space  & F(12,5)=(119,120,169)
 \end{align*}
Every adjacent pair corresponds to  an $\space n^{th}\space$ Pythagorean triple where $\space n\in\mathbb{N}.$
A: In set theory, it is common to define a function $f$ as any set of ordered pairs such that if $(a,b)\in f$, and $(a,c)\in f$, then $b=c$. Then, we can define a sequence as a function with a domain of $\Bbb N$. So the answer to your question is yes, we can think of elements of sequences as ordered pairs.
However, many mathematicians would feel uncomfortable with the idea that a sequence is literally just a set of ordered pairs satisfying certain properties. Arguably, the set theoretic construction of sequences only exists to demonstrate that they can be given a formal definition, meaning that we can feel confident that the everyday notion of sequences doesn't lead to any sort of contradiction. But in practice, it is seldom useful to think of mathematical objects as literally being the same as their set theoretic representations. For further discussion of this philosophical point, see here.
