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Binet Forms

The two Recurrence Sequences

$\displaystyle U_n$ $\textstyle =$ $\displaystyle mU_{n-1}+U_{n-2}$ (1)
$\displaystyle V_n$ $\textstyle =$ $\displaystyle mV_{n-1}+V_{n-2}$ (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
$\displaystyle U_n$ $\textstyle =$ $\displaystyle {\alpha^n-\beta^n\over \Delta}$ (3)
$\displaystyle V_n$ $\textstyle =$ $\displaystyle \alpha^n+\beta^n,$ (4)

$\displaystyle \Delta$ $\textstyle \equiv$ $\displaystyle \sqrt{m^2+4}$ (5)
$\displaystyle \alpha$ $\textstyle \equiv$ $\displaystyle {m+\Delta\over 2}$ (6)
$\displaystyle \beta$ $\textstyle \equiv$ $\displaystyle {m-\Delta\over 2}.$ (7)

A useful related identity is
\end{displaymath} (8)

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

See also Fibonacci Q-Matrix

© 1996-9 Eric W. Weisstein