I believe this to be among if not the MOST fascinating number!
What number, when multiplied by the same number minus 1 equals 1?
In other words, x*(x-1)=1
It turns out that through the use of the Pythagorean theorem, we get (1+ SQRT(5))/2=1.618...And thus 1.618... multiplied by 0.618... equals 1!
There is an amazing association with the Fibonacci Series:
1,1,2,3,5,8,13,21,35,...
In fact, the Fibonacci Series could be extended in the opposite direction:
...-35,21,-13,8,-5,3,-2,1,-1,0,1,1,2,3,5,8,13,21,35...
F(1)=1, F(2)=1, and with the extension, F(0)=0, F(-1)=-1, F(-2)=1, etc.
With the notation that the golden ratio = Phi,
F(n)=(1/sqrt(5))*Phi^n-(1/sqrt(5))*(1-Phi)^n
There are many ways of obtaining this. One way is from starting with the expression for Phi^n shown a little bit lower which I think is convoluted as it is in reverse order relative to the image displayed lower, but the one I find remarkably simple is the one put forth by Hall [note the corrections] at: Derivation for explicit formula for the Fibonacci Series. Here's a video using matrices (if you've been through linear algebra you may be able to follow, provided you retained what you learned - not all of us are so much into eigenvalues on an ongoing basis to know this second-hand: Derivative of the explicit formula for the Fibonacci Series using matrices. This page provides some other approach which is fascinating: Fibonacci Numbers Spelled Out.
An interesting formula using Pascal's triangle is:
F(n+1)=∑ from k=0 to roundup(n/2) of C(n−k,k)
Many more identities can be found at: Fibonacci Number
F(n) = RND((Phi^(n-1))/(3-Phi)); n>1 (independent derivation)
F(n) = round( Phi^n / √5 ) provided n ≥ 0 (simpler formula than mine as shown on A formula for nth Fibonacci number, also note the graphs at the bottom of this referenced page!)
I derived the following formula independently though I did see it published after the date of my discovery (I am not saying I was the first person as I would think someone else would have discovered this long before myself):
(Phi)^n=F(n)*Phi+F(n-1), and this works for all integers, noting the extension of values for the Fibonacci Series for all integers! This is an amazing discovery as any power of the golden ratio, even to a negative integral power is a linear combination of the golden ratio with integral coefficients! TRULY INCREDIBLE!
