Binet Form

First Form
The recursive sequence:
 * $U_n = m U_{n-1} + U_{n-2}$

where:

has the closed-form solution:
 * $U_n = \dfrac {\alpha^n - \beta^n} {\Delta}$

where:

Second Form
The recursive sequence:
 * $V_n = m V_{n-1} + V_{n-2}$

where:

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 $m$:
 * $U_{n-1} + U_{n+1} = V_n$