Fun with Fibonacci

[Allocthonous]

As I was lying in bed trying to fall back asleep this morning, I was amusing myself with some mental math, namely the Fibonacci sequence. Starting with 1, add the previous number to get the next number in the sequence. 1,1,2,3,5,8,13,21,34...

I was messing around a bit with the relationships between the numbers, and hit on something kind of cool. If you take any two consecutive numbers in the sequence, square them, and add them together, you'll get the number in the sequence found by adding the places of the consecutive numbers together. For example, sequence places 3 and 4: 2^2+3^2=13, which is 7th number in the sequence. For sequence places 4 and 5: 3^2+25^2=34, which is 9th in the sequence. This holds true for any pair of consecutive Fibonacci numbers, as far as I can tell. If you want the fancy formal notation of this, it would look something like:

For F_n=F_n-1+F_n-2 F₁=1, F₂=1,

F_n+(n+1)=(F_n)²+(F_n+1)²

(If I've done that wrong, please tell me. My math education has only touched on sequences and their notation very briefly.)

I have no idea why this happens, or if it's good for anything. I'm chalking it up to the golden ratio being spooky. I couldn't find anything on the wikipedia page for the Fibonacci sequence about this particular feature of it, but I'd be really shocked if it wasn't mentioned somewhere already.

Anyway, I thought this was neat and wanted to share. Hope somebody finds it interesting.

[Tullius]

It took me a bit of time to find the proof, but the German wikipedia page gave the right clue:

Fibonacci numbers satisfy the formula F_m+n = F_n+1 F_m + F_n F_m-1, of which your formula is the special case for n=n and m=n+1. The formula itself follows quite easily from the defining property F_n=F_n-1+F_n-2 via induction on m.