Preparing for Fibonacci Day: What Number Come Before the 1 and the 1?

The Fibonacci sequence “starts” with 1, and 1 (or some say 0, 1, and 1), and each number after that is obtained by adding the prior two numbers.  But is it possible that there are number that can come before the 1, and 1?

___, 1, 1

What number could go in the blank spot so that the first two number add up to the third number?  Something plus one equals one?

OK, I see a hand up in the back – just shout it out.

“Zero!”

Yep, 0 + 1 = 1.  What would be a number that I could add to 0 to get the first 1?

“One!”

1, 0, 1, 1, …

Let’s keep working backwards.  What number could I add to one in order to get 0?

“Negative one!”

That’s right – a negative one.

…, -1, 1, 0, 1, 1, 2, 3, 5, 8, …

So the answer is Yes – there are numbers that can come before the Fibonacci numbers – but what kind of pattern do they follow?

We could keep on going like this, but there is another way of getting the same answers.  Going forward we add two consecutive Fibonacci number to get the next number.  If we want to go backwards we subtract the two numbers to get the previous number.

1 – 1 = 0, 1 – 0 = 1, 0 – 1 = -1, 1 – (-1) = 2, -1 – 2 = -3, 2 – (-3) = 5, -3 – 5 = -8.  Now what do we have?

…, -8, 5, -3, 2, -1, 1, 0, 1, 1, 2, 3, 5, 8, …

Do you see a pattern?

Could you start with -8 and 5 and get the original Fibonacci sequence?  Would it work if you started with 8 and -5?  Are there numbers that come before the -8?

These numbers are known as the “NegaFibonacci” numbers.

David