The Fibonacci sequence is shown in red and defined by Binet's formula:

${\displaystyle f_{n}={\frac {1}{\sqrt {5}}}\left[\left({\frac {1+{\sqrt {5}}}{2}}\right)^{n}-\left({\frac {1-{\sqrt {5}}}{2}}\right)^{n}\right]=u_{n}-l_{n}~}$

Upper black sequence:

${\displaystyle u_{n}={\frac {1}{\sqrt {5}}}\left({\frac {1+{\sqrt {5}}}{2}}\right)^{n}}$

Lower black sequence (alternating):

${\displaystyle l_{n}={\frac {1}{\sqrt {5}}}\left({\frac {1-{\sqrt {5}}}{2}}\right)^{n}}$

 This plot was created with Ultiplot.

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