Fibonacci: Closed-Form solution

fib[n_]:=(GoldenRatio^n-(-GoldenRatio)^(-n))/Sqrt[5]

I’m going to round calculations, just to force Mathematica to give an integer (it wants to give the exact value):

fib[4]
Round[fib[Range[1,10]]]

Notice that “Fibonacci numbers” can be indexed backwards, to negative infinity! What does it mean to have negative pairs of rabbits?:)

Round[fib[Range[-10,10]]]

Now let’s look at that limit of ratios of successive Fibonacci numbers:

Limit[fib[n+1]/fib[n],n -> Infinity]

An even briefer closed-form solution

Interestingly enough, those inverse powers of the Golden ratio get small so fast that simply by throwing them away and rounding we get all the Fibonacci numbers:

fibPos[n_]:=Round[GoldenRatio^n/Sqrt[5]]

fibPos[5]
fibPos[Range[1,10]]

This proves that the Fibonacci numbers grow exponentially, like powers of the Golden ratio

But this approximation won’t produce the “negative Fibonacci numbers”:

fibPos[Range[-10,10]]

But it won’t produce the negative one:

fibPos[Range[-10,10]]

To get the negative one, round the other part of the original solution!

fibNeg[n_]:=Round[-(-GoldenRatio)^(-n)/Sqrt[5]]

fibNeg[Range[-10,10]]