Random idea – since we have a formula for the Fibonacci numbers, what if we plug in values that aren’t natural numbers?

Let’s start with negative numbers – you garner an interesting result when you plug those in! They are very similar to the standard Fibonacci numbers, with alternating positive and negative signs. If you start at 0, and instead count backwards (so you find the -1st, -2nd, -3rd, -4th and so on Fibonacci numbers), you get this:

0,1,-1,2,-3,5,-8,13,-21…

And so on.

What’s interesting here is that if you add the first “n” positive entries, and add it to the negative value right after the nth entry, it adds to 0. For example,

1,-1,2,-3,5,-8,13,-21

(The 0 has been removed for convenience.) Where 1+2+5+13=21, and 21-21=0.

Similarly, adding a sequence of the negatives to the positive value directly to their right gives 1.

1,-1,2,-3,5,-8,13,-21,34

-1-3-8-21+34=1

If we go back to our first sequence, where we add up the positive numbers, we see that we don’t actually need to use the negative Fibonacci numbers – all we need to do is say “Starting on the F1 (which is the first “1”), add every second Fibonacci number for “n” steps (this gets you to the 2n+1th Fibonacci number.) This number will be the same value as the Fibonacci number just to the right of the last term of the sum (the 2n+2nd Fibonacci number).

1,1,2,3,5,8,13,21

So this trick isn’t really related to negative Fibonacci at all! It just makes it a little harder to find whatever patterns there may be when negatives are involved.

PROOF OF THE ABOVE TRICK

What about real numbers? Non integer values? Now this is where things get interesting. The result of, for example, plugging in 1.5 is a complex number – 0.920442 + 0.217287i. If you were to do a 3d graph of the input (a real number) against the output (a complex number), you get this:

(For the nth Fibonacci numbers where n runs from 0 to 5). If you take out the input, and just graph the real part against the complex part (X and Y axes respectively) you get this:

(Who needs axis labels) If you graph the negative numbers (again, real part against the complex part), you get a nice shiny gold spiral:

(This isn’t actually a proper golden spiral, but it is related to the golden ratio)