Journal of Integer Sequences, Vol. 29 (2026), Article 26.4.4

Pairs of Intertwined Integer Sequences


Christian Kassel
Institut de Recherche Mathématique Avancée
Université de Strasbourg & CNRS
7 rue René-Descartes
67084 Strasbourg
France

Christophe Reutenauer
Mathématiques
Université du Québec à Montréal
CP 8888, succ. centre-ville
Montréal, PQ H3C 3P8
Canada

Abstract:

In previous work we computed the number $C_n(q)$ of ideals of codimension $n$ of the algebra $\mathbb{F}_q[x,y,x^{-1}, y^{-1}]$ of two-variable Laurent polynomials over a finite field  $\mathbb{F}_q$: it turned out that $C_n(q)$ is a palindromic polynomial of degree $2n$ in $q$, divisible by $(q-1)^2$. The quotient $P_n(q) = C_n(q)/ (q-1)^2$ is a palindromic polynomial of degree $2n-2$. For each $n\geq 1$ let  $\overline{P}_n(X) \in \mathbb{Z}[X]$ be the degree $n-1$ polynomial such that $\overline{P}_n(q+q^{-1}) = P_n(q)/q^{n-1}$. In this article we show that for every integer $N$ the integer value $\overline{P}_n(N)$ is close to the value at $N$ of the degree $n-1$ polynomial $F_{n-1}(X) = 1 + \sum_{k=1}^{n-1} \, \overline{T}_k(X)$, which is a sum of monic versions $\overline{T}_k(X)$ of Chebyshev polynomials of the first kind. We give a precise formula for  $\overline{P}_n(X)$ as a linear combination of the $F_k(X)$, each appearance of the latter being indexed by an odd divisor of $n$. As a consequence, $\overline{P}_n(X) = F_{n-1}(X)$ if and only if $n$ is a power of $2$. We exhibit similar formulas for $C_n(q)$.

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(Concerned with sequences A000096 A000203 A000215 A000668 A001834 A002325 A002654 A002878 A030221 A050415 A057733 A113063 A181703 A329156 A386706 A387017.)

Received September 19 2025; revised version received March 2 2026. Published in Journal of Integer Sequences, July 4 2026.

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