Fibonacci Number plus Arbitrary Function in terms of Fibonacci Numbers

Theorem
Let $\map f n$ and $\map g n$ be arbitrary arithmetic functions.

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

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

Let $\sequence {b_n}$ be the sequence defined as:
 * $b_n = \begin{cases}

0 & : n = 0 \\ 1 & : n = 1 \\ b_{n - 1} + b_{n - 2} + \map g {n - 2} & : n > 1 \end{cases}$

Let $\sequence {c_n}$ be the sequence defined as:
 * $c_n = \begin{cases}

0 & : n = 0 \\ 1 & : n = 1 \\ c_{n - 1} + c_{n - 2} + x \map f {n - 2} + y \map g {n - 2} & : n > 1 \end{cases}$

where $x$ and $y$ are arbitrary.

Then $\sequence {c_n}$ can be expressed in Fibonacci numbers as:
 * $c_n = x a_n + y b_n + \paren {1 - x - y} F_n$

Lemma
Hence also:


 * $b_n = F_n + \ds \sum_{k \mathop = 0}^{n - 2} F_{n - k - 1} \map g k$

Thus: