Wir betrachten die reelle Folge \( \left(t_{n}\right)_{n \in \mathbb{N}_{0}} \), welche wie folgt definiert ist:
$$ t_{n}:=\left\{\begin{array}{ll} 0 & \text { falls } n=0, \\ 2 & \text { falls } n=1, \\ 3 & \text { falls } n=2, \\ 2 t_{n-1}+t_{n-2}-2 t_{n-3} & \text { falls } n \geqslant 3 . \end{array}\right. $$
Bestimmen Sie einen geschlossenen Ausdruck für \( \left(t_{n}\right)_{n \in \mathbb{N}_{0}} \) durch Anwenden der Theorie von Theorem :
Consider the linear inhomogeneous recurrence relation
$$ a_{0} t_{n}+a_{1} t_{n-1}+\cdots+a_{k} t_{n-k}=\sum \limits_{i=1}^{m} b_{i}^{n} \cdot p_{i}(n) $$
where \( b_{i} \) are constants and \( p_{i} \) are polynomials in \( n \) of degree \( d_{i} \in \mathbb{N}_{0} \), for \( m \in \mathbb{N}_{0} \), and suppose that its characteristic equation
$$ \left(a_{0} x^{k}+a_{1} x^{k-1}+\cdots+a_{k}\right) \cdot \prod \limits_{i=1}^{m}\left(x-b_{i}\right)^{d_{i}+1}=0 $$
has s distinct roots \( r_{1}, \ldots, r_{s} \) of multiplicities \( m_{1}, \ldots, m_{s} \) such that all roots are real numbers. Then the general solution of the recurrence relation is given by
$$ t_{n}=\sum \limits_{i=1}^{s} \sum \limits_{j=0}^{m_{i}-1} c_{i j} \cdot n^{j} \cdot r_{i}^{n} $$
for constants \( c_{i j} \in \mathbb{R} \)
