fibonacci_sequence []

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Recurrence relation $$F_0 = 0, \quad F_1 = 1$$

and $$F_n=F_{n-1} + F_{n-2}$$ for $ n > 1 $

The first 20 Fibonacci numbers $F_n$ are:

$ F_0 $	$ F_1 $	$ F_2 $	$ F_3 $	$ F_4 $	$ F_5 $	$ F_6 $	$ F_7 $	$ F_8 $	$ F_9 $	$ F_{10} $	$ F_{11} $	$ F_{12} $	$ F_{13} $	$ F_{14} $	$ F_{15} $	$ F_{16} $	$ F_{17} $	$ F_{18} $	$ F_{19} $
0	1	1	2	3	5	8	13	21	34	55	89	144	233	377	610	987	1597	2584	4181

The sequence is (0, 1, 1, 2, 3, 5, 8, 13, … )

$$ {F_{k+2} \choose F_{k+1}} = \begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix} {F_{k+1} \choose F_{k}} $$

$$ \vec F_{k+1} = \mathbf{A} \vec F_{k} $$

$$ \vec F_n = \mathbf{A}^n \vec F_0 $$