The Fibonacci sequence.

This looks a lot better. You saw nothing.

It's a Fibonacci sequence.

It is the Fibonacci sequence.

It's a great way to teach recursion.

Your grandfather's Fibonacci sequence.

So here I'm (Applause)

The rest takes care of itself.

Or why not go back one more to 1, and so on?

13.

You might have noticed that up here on the right, I made a very simple chart to try and explain how Fibonacci behaves to myself.

This is certainly true.

Look familiar? 8 and 13 are both numbers in the 'Fibonacci series.'

COUNTS TO

Maybe plants just like these numbers, though.

The leaves are hard to keep track of.

It's seems pretty cosmic and wondrous, but the cool thing about the Fibonacci series and spiral is not that it's this big, complicated, mystical, magical, super math thing, beyond the comprehension of our puny human minds, that shows up mysteriously everywhere.

So just as we applied memoization to Fibonacci, we're going to apply memoization to our attempts to parse a grammar.

And when I'm going to figure these out, I don't have to do a huge amount of work.

The ratios of consecutive numbers also approach phi.

1, 3, 4, 7, 11, 18. is kind of like the Fibonacci series, because 1 3 is 4 3 4 is 7 and so on

And now that we've seen all the theory behind it, we're going to code it up in Python together.

Or the Brussels sprouts on this stalk are a beautiful, delicious 3 and 5.

The rest happens auto math ically.

Either way is a perfectly good series, and it's got another similarity with the Fibonacci series.

In an incredible surprise move, it actually shows up a lot in nature for example, in the patterns of seeds in a sunflower or in the whirls in a clamshell or in yellow chamomile plants or all that good stuff.

Just additional those 2 chart cells together and get the answer.

Most of you probably ran into the Fibonacci sequence of numbers, named after filius Bonacci in a previous computer science class.

Number of cols

Number of columns

Another kind of statement is the return statement.

In other words, the number of columns in this matrix, so it's the number of n columns.

When we were trying to compute Fibonacci, the Nth position in the chart corresponded to the Nth Fibonacci number.

And you can keep going! try fibonacci of 3! it'll work! because we know fibonacci of 0,1,or 2 works cuz fibonacci of 3 will boil down to fibonacci of 2 plus fibonacci of 1. we know that fibonacci of 2 is 1, fibonacci of 1 is 1 1 1 is 2 so it'll just keep working. now we could try it out

Fibonacci numbers show up occasionally in nature.

The fibonacci sequence makes numerous appearances throughout the natural world.

This is the most practical application in the world.

And once you're memorizing double digits, you might as well know 21, 34, 55, 89 so that whenever someone turns a Fibonacci number, you can say, Happy Fib birthday!

Number of columns

Let's just go check.

This one was 8 and 13.

That's going to be something like document.write(55) could be more complicated, calling Fibonacci functions doing arbitrary computation, in fact.

So there are Fibonacci numbers in pine cones.

Number of thumbnails columns

Fibonacci is even in the arrangement of the petals on this rose.