# Sequence Notation

I have come across the following in the book "Principles of Program Analysis" by Nielson, Nielson and Hankin to represent a sequence and I am unsure of what its constituent parts mean. Obviously $\vec{\emptyset}$ is the empty vector and F is a function but what are the sub / superscript n's for and why does it represent a sequence?

$$(F^n(\vec{\emptyset}))_n$$

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It would help to know the exact book (title and author). Normally (I've seen) when specifying a sequence, $({f})_n$ is used merely to specify it's a sequence on $\mathbb{N}$. In some books the parentheses are replaced by {} brackets, so I guess the subscript $n$ distinguishes it from a normal set. – Tyler Mar 5 '12 at 15:52
This reminds me of $F:\emptyset \to \emptyset$. Is it continuous? Open? Bounded? Perhaps in your case, the $\emptyset$ is just to show off that it's independent of any particular vector? I'm not familiar with that notation. – mixedmath Mar 5 '12 at 16:19
Updated the question. – Jonathan Evans Mar 5 '12 at 19:09

## 1 Answer

I am assuming that $F$ is a function from some set back to itself, so that $F^n$ stands for $F\circ F\circ\dots\circ F$ , the n-fold composition, and that your $\emptyset$ thing is the $0$ vector. The subscript $n$ indicates that $F^n(0)$ is the $n$-th term of the sequence. (Under these assumptions, if $F$ linear then the sequence is trivial)

In general, a sequence in a set $A$ is just a function $s:\mathbb{N}\rightarrow A$, where $s(n)$ is viewed as the n-th term of the sequence. Often we see $\{s(n)\}_n$ or $\{s(n)\}_{n\in\mathbb{N}}$ denoting the image of $s$, indicating that this image is parametrized by $n\in\mathbb{N}$. In your case, $s(n)=F^n(0)$.

As another example, if we want a concise way of writing the sequence of rational numbers $\frac{1}{n}$ as $n$ ranges over $\mathbb{N}$, we could write $\{\frac{1}{n}\}_n$

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