Fibonacci numbers and the Golden Ratio are NOT the same thing…but you knew that :)

Closed-form expression

Like every sequence defined by linear recurrence,[16] the Fibonacci numbers have a closed-form solution. It has become very well known as Binet‘s formula, even though it was already known by Abraham de Moivre:[17]

Fleft(nright) = {{varphi^n-(1-varphi)^n} over {sqrt 5}}={{varphi^n-(-1/varphi)^{n}} over {sqrt 5}}, ,


varphi = frac{1 + sqrt{5}}{2} approx 1.61803,39887dots,

is the golden ratio (sequence A001622 in OEIS).



follows from the defining equation above.

The Fibonacci recursion


is similar to the defining equation of the golden ratio in the form


which is also known as the generating polynomial of the recursion.

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