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.

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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.Terminal Output

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

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