## Symbolic Closed-Form Fibonacci

Let $V := \{(a_j)_{j\in\mathbb{N}}\subset\mathbb{C}|a_n=a_{n-1}+a_{n-2}\forall n>1\}$ be the two-dimensional complex vector space of sequences adhering to the Fibonacci recurrence relation with basis $B := ((0,1,\dots), (1,0,\dots))$.
Let furthermore $f: V\to V, (a_j)_{j\in\mathbb{N}}\mapsto(a_{j+1})_{j\in\mathbb{N}}$ be the sequence shift endomorphism represented by the transformation matrix

$A := M^B_B(f) = \begin{pmatrix}1&1\\1&0\end{pmatrix}$.

By iteratively applying the sequence shift a closed-form solution for the standard Fibonacci sequence follows.

## φ Generator

This program generates φ, also called the golden ratio. It creates the fibonacci sequence $\big\{1, 1, 2, 3, 5, 8, 13, \dots\big\}$ and divides the newly generated number by the last one. In theory this program would generate exactly φ.

#### Fibonacci sequence

$\text{Start: } x_1 = 1 \text{ and } x_2 = 1$
$\text{Generation: } x_n = x_{n-1} + x_{n-2}$

$1 + 1 = 2$
$1 + 2 = 3$
$2 + 3 = 5$
$3 + 5 = 8$
$5 + 8 = 13$
$8 + 13 = \dots$

#### The golden ratio (φ)

$\phi \text{ is the ratio between } x_n \text{ and } x_{n-1}\text{.}$
$\phi = \frac{x_n}{x_{n-1}}$

# Python 2.7.7 Code
# Jonathan Frech 6th of April, 2015