Generalized Fibonacci sequence

A generalization of the Fibonacci sequence is either an extension of the Fibonacci sequence to larger areas of definition than the natural numbers or a generalization of the law of formation.

Extension to larger areas of definition

Extension to all whole numbers

If you reverse the law of formation of the Fibonacci sequences, you get

${\displaystyle f_{n-2}=f_{n}-f_{n-1}}$.

With this formula you can recursively calculate Fibonacci numbers to negative whole numbers. The Moivre-Binet formula also applies to negative whole numbers: The following applies to the golden section : ${\displaystyle \varphi }$

${\displaystyle 1+{\frac {1}{\varphi }}=\varphi \Leftrightarrow {\frac {1}{\varphi }}=\varphi -1\Leftrightarrow 1-\varphi -1={\frac {1}{1-\varphi }}}$

If one sets , it follows from ${\displaystyle \psi :=1-\varphi }$

${\displaystyle {\tfrac {\varphi ^{0}-\psi ^{0}}{\sqrt {5}}}=0=f_{0}}$, ${\displaystyle {\tfrac {\varphi ^{1}-\psi ^{1}}{\sqrt {5}}}=1=f_{1}}$

and

${\displaystyle f_{-(n+1)}=f_{-(n-1)-2}=f_{-(n-1)}-f_{-n}}$

${\displaystyle ={\frac {\varphi ^{-n+1}-\psi ^{-n+1}}{\sqrt {5}}}-{\frac {\varphi ^{-n}-\psi ^{-n}}{\sqrt {5}}}={\frac {{\frac {\varphi -1}{\varphi ^{n}}}-{\frac {\psi -1}{\psi ^{n}}}}{\sqrt {5}}}={\frac {\varphi ^{-(n+1)}-\psi ^{-(n+1)}}{\sqrt {5}}}}$.

The induction conclusion results

${\displaystyle \forall n\in \mathbb {N} _{0}:f_{-n}={\frac {\varphi ^{-n}-\psi ^{-n}}{\sqrt {5}}}}$,

so that finally the formula of Moivre-Binet

${\displaystyle \forall n\in \mathbb {Z} :f_{n}={\frac {\varphi ^{n}-\psi ^{n}}{\sqrt {5}}}}$

holds for all integers.

Extension to all complex numbers

The closed form for the -th Fibonacci number is for whole numbers (see above ): ${\displaystyle n}$

${\displaystyle f_{n}={\frac {\Phi ^{n}-(1-\Phi )^{n}}{\sqrt {5}}},\qquad n\in \mathbb {Z} }$,

where is the golden ratio . The following equation applies to the golden section : ${\displaystyle \Phi }$ ${\displaystyle \Phi }$

${\displaystyle 1+{\frac {1}{\Phi }}=\Phi \Leftrightarrow (-1){\frac {1}{\Phi }}=1-\Phi \Rightarrow (1-\Phi )^{n}=(-1)^{n}\left({\frac {1}{\Phi }}\right)^{n}=(-1)^{n}\Phi ^{-n}}$

If it is an integer, however: ${\displaystyle n}$

${\displaystyle (-1)^{n}=\cos(n\pi )}$

Therefore the continuous and analytic function

${\displaystyle \operatorname {Fib} (x)={\frac {\Phi ^{x}-\cos(x\pi )\Phi ^{-x}}{\sqrt {5}}}}$

a continuation of the Fibonacci numbers on the complex numbers.

Generalization of the Education Act

Lucas episode

The Fibonacci sequence is a special case of the Lucas sequence .

Consequences with a similar education law

Sequences in the complex numbers

Let be a sequence in that for by the recursive law of formation ${\displaystyle (g_{n})_{n\in \mathbb {N} _{0}}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle n\geq 0}$

${\displaystyle g_{n+2}=g_{n+1}+g_{n}}$

is defined, such a sequence is a generalization of the Fibonacci sequence, since it arises when one puts and . There is a closed expression for the -th member of this sequence: ${\displaystyle g_{0}=0}$ ${\displaystyle g_{1}=1}$ ${\displaystyle n}$

${\displaystyle g_{n}=f_{n}\cdot g_{1}+f_{n-1}\cdot g_{0},\quad n\geq 0}$,

wherein the th Fibonacci number. This follows from complete induction with induction beginning ${\displaystyle f_{n}}$ ${\displaystyle n}$

${\displaystyle g_{0}=0\cdot g_{1}+1\cdot g_{0}=f_{0}\cdot g_{1}+f_{-1}\cdot g_{0}}$

and induction step

${\displaystyle g_{n}=g_{n-1}+g_{n-2}=g_{1}\cdot (f_{n-1}+f_{n-2})+g_{0}\cdot (f_{n-2}+f_{n-3})=g_{1}\cdot f_{n}+g_{0}\cdot f_{n-1}}$

Sequences of vectors

If is a vector space and are , a sequence of vectors can be recursively defined by ${\displaystyle V}$ ${\displaystyle g_{0},g_{1}\in V}$ ${\displaystyle (g_{n})_{n\in \mathbb {N} _{0}}}$ ${\displaystyle g_{n}\in V}$

${\displaystyle g_{n+2}=g_{n+1}+g_{n},\quad n\geq 0}$.

As above , the formula then applies

${\displaystyle g_{n}=f_{n}\cdot g_{1}+f_{n-1}\cdot g_{0},\quad n\geq 0}$.

Vector space of the Fibonacci sequences

Because of the equation

${\displaystyle g_{n}=f_{n}\cdot g_{1}+f_{n-1}\cdot g_{0},\quad n\geq 0}$

the set of sequences with can also be understood as an infinite-dimensional vector space, for which and form the basis. ${\displaystyle (g_{n})_{n\in \mathbb {N} _{0}}}$ ${\displaystyle g_{n+2}=g_{n+1}+g_{n}}$ ${\displaystyle \mathbb {C} }$ ${\displaystyle (f_{n})_{n\in \mathbb {N} _{0}}}$ ${\displaystyle (f_{n-1})_{n\in \mathbb {N} _{0}}}$

Individual evidence

1. ↑ Harry J. Smith: What is a Fibonacci Number? In: geocities.com. October 20, 2004, archived from the original on 20091027103713 ; accessed on January 13, 2015 .