  Let $C_n(q)$ be the number of ideals of codimension $n$ of
$\mathbb{F}_q\left[x, y, x^{-1}, y^{-1} \right]$, 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 $n \geq 1$.
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
  $$ F(t) = \sum_{n \geq 1} f_{2n}\,t^n. $$
  We will prove that for each $n \geq 1$,
  $$ C_n\left( \frac{3+\sqrt{5}}{2}\right) = \lambda_n \, \left(f_{2n}
\frac{3+\sqrt{5}}{2} - f_{2n-2} \right) , $$ where the integers $\lambda_n \geq
0$ are given by the following generating function
  $$ \prod_{m \geq 1} \left(1+F\left( t^m\right)\right) = 1 + \sum_{n \geq 1}
\lambda_n\,t^n. $$
