# Binet Form

## Theorem

### First Form

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

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

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

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 $m$:

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

## Source of Name

This entry was named for Jacques Philippe Marie Binet.