Binet Forms

The two Recurrence Sequences

$$U_n = mU_{n-1}+U_{n-2}$$ (1)

$$V_n = mV_{n-1}+V_{n-2}$$ (2)

with $U_0=0$, $U_1=1$ and $V_0=2$, $V_1=m$, can be solved for the individual $U_n$ and $V_n$. They are given by

$$U_n = {\alpha^n-\beta^n\over \Delta}$$ (3)

$$V_n = \alpha^n+\beta^n,$$ (4)

where

$$\Delta \equiv \sqrt{m^2+4}$$ (5)

$$\alpha \equiv {m+\Delta\over 2}$$ (6)

$$\beta \equiv {m-\Delta\over 2}.$$ (7)

A useful related identity is

$$U_{n-1}+U_{n+1}=V_n.$$ (8)

Binet's Formula is a special case of the Binet form for $U_n$ corresponding to $m=1$.

See also Fibonacci Q-Matrix