Binet Form

Theorem

First Form

The recursive sequence:

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

where:

	\(\displaystyle \)	\(\displaystyle U_0\)	\(=\)	\(\displaystyle 0\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle U_1\)	\(=\)	\(\displaystyle 1\)	\(\displaystyle \)

has the closed-form solution:

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

where:

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

Second Form

The recursive sequence:

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

where:

	\(\displaystyle \)	\(\displaystyle V_0\)	\(=\)	\(\displaystyle 2\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle V_1\)	\(=\)	\(\displaystyle m\)	\(\displaystyle \)

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.

Sources

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