Fibonacci Number plus Binomial Coefficient in terms of Fibonacci Numbers

Theorem
Let $m \in \Z_{>0}$ be a positive integer.

Let $\left\langle{a_n}\right\rangle$ be the sequence defined as:
 * $a_n = \begin{cases}

0 & : n = 0 \\ 1 & : n = 1 \\ a_{n - 2} + a_{n - 1} + \dbinom {n - 2} m & : n > 1 \end{cases}$

where $\dbinom {n - 2} m$ denotes a binonial coefficient.

Then $\left\langle{a_n}\right\rangle$ can be expressed in Fibonacci numbers as:
 * $a_n = F_{m + 1} F_{n - 1} + \left({F_{m + 2} + 1}\right) F_n - \displaystyle \sum_{k \mathop = 0}^m \dbinom {n + m - k} k$