A relationship between the ideals of ${\mathbb{F}}_q[x,y,x^{-1},y^{-1}]$ and the Fibonacci numbers

Let $C_n(q)$ be the number of ideals of codimension $n$ of ${\mathbb{F}}_q[x,y,x^{-1},y^{-1}]$, where ${\mathbb{F}}_q$ is the finite field with $q$ elements. Kassel and Reutenauer [KasselReutenauer2015A] proved that $C_n(q)$ is a polynomial in $q$ for any fixed value of $n\ge 1$. For $q=\frac{3+\sqrt{5}}{2}$, this combinatorial interpretation of $C_n(q)$ is lost. Nevertheless, an unexpected connexion with Fibonacci numbers appears. Let $f_n$ be the $n$-th Fibonacci number (following the convention $f_0=0$, $f_1=1$). Define the series $$F(t)=\sum _{n\ge 1}{f}_{2n}{t}^n\mathrm{.}$$ We will prove that for each $n\ge 1$, $$C_n\left(\frac{3+\sqrt{5}}{2}\right)={\lambda }_n(f_{2n}\frac{3+\sqrt{5}}{2}-f_{2n-2}),$$ where the integers $\lambda_n \geq 0$aregivenbythefollowinggeneratingfunction$$ \prod_{m \geq 1} \left(1+F\left( t^m\right)\right) = 1 + \sum_{n \geq 1} \lambda_n,t^n. $$