Sequences, Math I have read definitions of sequence, different authors define it somewhat different. I know that a sequence is special type of function and I have read that if numbers follow a special pattern then they are called a sequence but I watched a lecture on YouTube where the teacher said any list of numbers is a sequence like $-1,2,3,4,-2,0,5,7,-7,-2$ is a sequence which obviously don't have a special pattern which caused my confusion because there is no such particular explanation of these numbers like Fibonacci sequence or the sequence of digits in the decimal expansion of $\pi$, or the sequence of digits after the decimal point when $1$ is divided by $3$ i.e $3,3,3,3,\ldots$. Please clear my confusion.
 A: A sequence in a set $X$ is a function $x: \mathbb{N}\rightarrow X$ for which we denote $x(n)$ by $x_n$. Furthermore, if $X=\Bbb{R}$, then we can call these real valued sequences. We then also write $(x_n)=(x_1,x_2,\ldots, x_n, \ldots, )$. 
There need not be any pattern at all. However, there are a lot of special sequences, for e.g. you might be able to see a pattern, for instance in $(0,-1,1,-2,2,-3,3,-4,4,\ldots )$ and even more special are convergent sequences, like $(1,\frac{1}{2},\frac{1}{3}, \ldots, \frac{1}{n}, \ldots )$.
The Fibonacci sequence is a sequence defined by a recurrence relation, i.e. by letting $F_1=1$, $F_2=2$ and $F_n = F_{n-1}+F_{n-2}$ for all $n \geq 2$. There are many ways to generate sequences, but they are all functions.
Also note that the digits of $\pi$ forms a sequence, but there is no general pattern. This will remain true for any transcendental number as well.
A: A sequence is a function whose domain is the set of natural numbers or a
subset of the natural numbers. 
Intuitively, a sequence is just an ordered list of (possibly infinitely many) numbers. 
The numbers need not to follow a special pattern.
A sequence may be finite or infinite.
A: A sequence is, quite literally, just a `list of things' and you are right that there does not have to be any pattern to the list. It just so happens that many sequences do have some pattern or some additional property that makes them interesting to study. To make an analogy, the words authority and lokwandias are both words because they are both strings of letters, but the word authority is the only one which has a meaning to us. It does not mean that lokwandias is not a word, simply that it is meaningless.
Similarly, the sequence $1,6,54,245,54,54,623,1,1,6,\ldots$ and the sequence $1,4,7,10,13,16,\ldots$ are both sequences, but only the second one seems meaningful because we can see a relationship between the terms. Don't get me wrong - sequences of random-looking numbers can be interesting! My point is that they are still sequences even though they seem patternless.
The other answers in this thread give a more rigorous notion of what a sequence is so I won't repeat it here.
A: A sequence of values from any set $S$ is a function from the natural numbers $\mathbb N=\{1,2, ...\}$ to $S$.
We usually write this as $\{x_1,x_2,...\}$ where it is understood that $x_i$ is the value of the function on $i$.
There is no special pattern involved for a sequence but most interesting sequences will have some pattern. To see why, think about how you would describe one.
