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Mathematics

(People have already mentioned many of the great properties of this sequence at the time of answering. These are a couple of good ones that were missed.)

Combinatorics

The Fibonacci sequence is the number of strings of characters of length $n$, containing only a and b , where two a 's cannot be consecutive. For $n=4$ there are $8$ strings for example:

abab abba abbb baba babb bbab bbba bbbb

Relation

Another nice factoid is that $F_{n-1} F_{n+1} = F_n^2 + (-1)^n$.

This relation, and lots of other nice ones, including the limiting behavior, follow from the closed form $$ F_n = \phi_1^n + \phi_2^n $$ where $\phi_1$ and $\phi_2$ are the two solutions to $x^2 - x - 1 = 0$ (and the positive one of these is the usual 'golden ratio' $\phi$).

Solving the recursion to find this closed form is neat and not too complicated, although it does require fluency with algebraic manipulation, and some tricky concepts. If you can do this, you can see why all Fibonacci sequences, regardless of the first two numbers, have the same limiting ratio between two adjacent terms, and how all the closed forms are related.

If you don't want to find the closed form, you can probably still obtain the relation, just using the recursive definition.

Phi

The irrational number which can be least well approximated by fractions of a given size of denominator, is $\phi$. The best approximations at any size are given by $\frac{F_n}{F_{n-1}}$.

For most irrational numbers, if you try to find fractions which are close to that number, but whose denominator is not too big, you occasionally stumble across fractions which are much close than the 'average'. Eg $\pi$ can be written approximately as $\frac{3}{1}$, $\frac{22}{7}$, and then $\frac{355}{113}$. The last one is much, much better than you would hope for - the error is one over 5 million whereas you should statistically only expect an error of one in a few tens of thousands, for a denominator of that size.

With $\phi$ you never 'get lucky' - fractional approximations, even the best possible fractional approximations, always fall roughly the expected distance away from $\phi$. They also alternate between too big and too small.

To fully understand the math behind this, you have to learn about continued fractions, which are extremely interesting, but probably off-topic for your class. The math is not very difficult, but it's not the kind of thing you can explain in a 10 minute parenthesis.

Pseudo-mathematics

A lot (perhaps most) of what is claimed about the Fibonacci sequence and the (very closely related) Golden Ratio is bogus. A nice paper is here: http://community.dur.ac.uk/bob.johnson/fibonacci/miscons.pdf

Some pointers:

Natural world

The Fibonacci sequence has a pretend real-world justification, in terms of rabbits reproducing. No-one sensible has ever claimed that these number were observed in real rabbit populations.

The Fibonacci sequence does appear in some plant physiology, notably numbers of branches, petals and seeds for certain plants. There are some good reasons for this: some plants branch in simple ways which are analogous to the 'rabbit family trees'.

Some of these patterns also involve the 'Fibonacci spiral', eg sunflower heads. This has a slightly more complex mathematical justification, which is not at all controversial.

Lots of people have found Fibonacci numbers, the golden ratio and Fibonacci spirals in many, many natural objects. Most of these claims are either coincidence or deliberate (perhaps unconscious) mismeasurement (unless the universe works very, very differently to the way most scientists think it does).

Examining 'fringe' claims

Some of these people are just on the wrong track, or doing something of interest to them which isn't quite mathematics or science, some of them are probably insane. It's often hard to tell papers written by these people apart from serious writing about the links between mathematics and say, zoology.

Some of them might be right, and have spotted observed something about the mathematics of say, a zebra's stripes, that wasn't previously known to professionals. What is very unlikely is that all of these claims are true.

The Golden ratio is probably the worst offender. If you make enough different measurements of parts of an object, you are pretty certain to find close approximations to any ratio you care to choose.

Finding small values from the Fibonacci sequence (such as 5 or 8) is another common trick.

Some writing which purports to find several occurrences of $\phi$ or the Fibonacci sequence in a given natural object, actually finds one such occurrence, and then transforms it in various simple mathematical ways and claims that these are independent observations.

The ratio of the side to the diagonal of a pentagon is $\phi$. Natural objects with fivefold symmetry (some fruit, starfish) will probably contain the golden ratio without this meaning much, except that they have fivefold symmetry.

Art and culture

Some historical and contemporary architects, artists and designers have intentionally used $\phi$ or the Fibonacci sequence. The most famous example is the Parthenon in Athens.

Some of these people believed that these ratios and numbers were specially beautiful, some that they had some kind of mystic power, and some were playing around with interesting math.

Some of the intentional use by artists might have degenerated into the belief that the numbers are present in physical objects. For example, artists might have chosen to use $\phi$ for the ratio of navel height to height when drawing figures, and this might have led people later to believe that this exact ratio actually occurs in all humans.

As with natural objects, it has also been claimed that $\phi$ and Fibonacci appear in almost all buildings, paintings, sculptures and music in various disguised or non-disguised forms. Much of the 'evidence' of this works the same way as with natural objects - measure many different parts of a painting, and work out all the ratios until you find the ones you want.

There is no evidence that our brain is wired to select these patterns unconsciously or to find them especially beautiful.

Nor is there a scientific consensus that $\phi$ and Fibonacci are ubiquitous in all human endeavour at all times in history because of divine inspiration, secret knowledge handed down by aliens, or an extremely wide-reaching conspiracy.

If you discuss this with your students you will probably hear a lot of the bogus claims above which have passed in the popular consciousness, partly as a result of 'The Da Vinci Code'. I think examining these is great math. Depending on how focussed on your syllabus you have to be, you might be able to discuss some or all of the following with your class:

How accurate does the ratio between a person's height and the height of their belly button have to be, before we can be sure that it is $1.618$ and not $1.638$ or $1.5922$? Is the related claim that the height of the belly button and the vertical distance between the top of the head and the belly button are in the same ratio further evidence, or is it redundant? How many people should we measure? How many daisies should we count petals of to decide if only Fibonacci numbers appear, and how close to the exact numbers do we need to be? What signs within a math paper and from its context should we examine to decide if it can be trusted? Is the distinction between mainstream math and pseudo-mathematics valid or is pseudo-mathematics just mathematics by people outside the mainstream?

You might or might not be interested in talking to your class about this. I had some success when students in a Calc I class asked me about appearances of $\pi$ in Egyptian pyramids. However even if you don't want to go into it, you should say just enough when talking about this topic that your students understand that not every claim they might read is reliable.