# Binet Form

From ProofWiki

## Contents

## Theorem

### First Form

The recursive sequence:

- $U_n = m U_{n-1} + U_{n-2}$

where:

\(\displaystyle U_0\) | \(=\) | \(\displaystyle 0\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle U_1\) | \(=\) | \(\displaystyle 1\) | $\quad$ | $\quad$ |

has the closed-form solution:

- $U_n = \dfrac {\alpha^n - \beta^n} {\Delta}$

where:

\(\displaystyle \Delta\) | \(=\) | \(\displaystyle \sqrt {m^2 + 4}\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \alpha\) | \(=\) | \(\displaystyle \frac {m + \Delta} 2\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle \beta\) | \(=\) | \(\displaystyle \frac {m - \Delta} 2\) | $\quad$ | $\quad$ |

### Second Form

The recursive sequence:

- $V_n = m V_{n-1} + V_{n-2}$

where:

\(\displaystyle V_0\) | \(=\) | \(\displaystyle 2\) | $\quad$ | $\quad$ | |||||||||

\(\displaystyle V_1\) | \(=\) | \(\displaystyle m\) | $\quad$ | $\quad$ |

has the closed-form solution:

- $V_n = \alpha^n + \beta^n$

where $\Delta, \alpha, \beta$ are as for the first form.

## Relation Between First and Second Form

For any given value of $m$:

- $U_{n-1} + U_{n+1} = V_n$

## Proof

## Source of Name

This entry was named for Jacques Philippe Marie Binet.

## Sources

- Weisstein, Eric W. "Binet Forms." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/BinetForms.html