Binet Form

Theorem

First Form

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

where:

 $\displaystyle U_0$ $=$ $\displaystyle 0$ $\displaystyle U_1$ $=$ $\displaystyle 1$

has the closed-form solution:

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

where:

 $\displaystyle \Delta$ $=$ $\displaystyle \sqrt {m^2 + 4}$ $\displaystyle \alpha$ $=$ $\displaystyle \frac {m + \Delta} 2$ $\displaystyle \beta$ $=$ $\displaystyle \frac {m - \Delta} 2$

Second Form

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

where:

 $\displaystyle V_0$ $=$ $\displaystyle 2$ $\displaystyle V_1$ $=$ $\displaystyle m$

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.