Binet Form

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

where:

$$ $$

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

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